Use the property Log b (xy) = Log b (x) + Log b(y) to expand: Log 3 (5x)

log ⁡ 2 ( 5 ) + log ⁡ 2 ( x ) \log_2\left(5\right)\ +\ \log_2\left(x\right) lo g 2 ( 5 ) + lo g 2 ( x )

log ⁡ 2 ( 5 ) × log ⁡ 2 ( x ) \log_2\left(5\right)\ \times\ \log_2\left(x\right) lo g 2 ( 5 ) × lo g 2 ( x )

log ⁡ 2 ( 5 ) − log ⁡ 2 ( x ) \log_2\left(5\right)\ -\ \log_2\left(x\right) lo g 2 ( 5 ) − lo g 2 ( x )

log ⁡ 2 ( 5 + x ) \log_2\left(5\ +\ x\right)\ lo g 2 ( 5 + x )

Use the property log ⁡ b ( x y ) = log ⁡ b ( x ) − log ⁡ b ( y ) \log_b\left(\frac{x}{y}\right)\ =\ \log_b\left(x\right)\ -\ \log_b\left(y\right) lo g b ( y x ) = lo g b ( x ) − lo g b ( y ) to expand log ⁡ 2 ( x 4 ) \log_2\left(\frac{x}{4}\right)\ lo g 2 ( 4 x )

log ⁡ 2 ( x ) − log ⁡ 2 ( 4 ) \log_2\left(x\right)\ -\ \log_2\left(4\right) lo g 2 ( x ) − lo g 2 ( 4 )

log ⁡ 2 ( x ) × log ⁡ 2 ( 4 ) \log_2\left(x\right)\ \times\ \log_2\left(4\right) lo g 2 ( x ) × lo g 2 ( 4 )

log ⁡ 2 ( x ) + log ⁡ 2 ( 4 ) \log_2\left(x\right)\ +\ \log_2\left(4\right) lo g 2 ( x ) + lo g 2 ( 4 )

log ⁡ 2 ( 5 ) − log ⁡ 2 ( x ) \log_2\left(5\right)\ -\ \log_2\left(x\right)\ lo g 2 ( 5 ) − lo g 2 ( x )

Use the change of base property log ⁡ b ( x ) = log ⁡ a ( x ) log ⁡ a ( b ) \log_b\left(x\right)\ =\ \frac{\log_a\left(x\right)}{\log_a\left(b\right)}\ lo g b ( x ) = lo g a ( b ) lo g a ( x ) to write log ⁡ 2 ( 16 ) \log_2\left(16\right)\ lo g 2 ( 1 6 ) with the common base

log ⁡ ( 16 ) log ⁡ ( 2 ) \frac{\log\left(16\right)}{\log\left(2\right)} lo g ( 2 ) lo g ( 1 6 )

log ⁡ ( 8 ) \log\left(8\right) lo g ( 8 )

log ⁡ ( 2 ) log ⁡ ( 16 ) \frac{\log\left(2\right)}{\log\left(16\right)} lo g ( 1 6 ) lo g ( 2 )

log ⁡ ( 16 ) + log ⁡ ( 2 ) \log\left(16\right)\ +\ \log\left(2\right)\ lo g ( 1 6 ) + lo g ( 2 )

Expand log ⁡ 4 ( 6 y ) \log_4\left(6y\right) lo g 4 ( 6 y )

log ⁡ 4 ( 6 ) + log ⁡ 4 ( y ) \log_4\left(6\right)\ +\ \log_4\left(y\right) lo g 4 ( 6 ) + lo g 4 ( y )

log ⁡ 4 ( 6 ) − log ⁡ 4 ( y ) \log_4\left(6\right)\ -\ \log_4\left(y\right) lo g 4 ( 6 ) − lo g 4 ( y )

y log ⁡ 4 ( 6 ) y\log_4\left(6\right)\ y lo g 4 ( 6 )

log ⁡ 4 ( 6 y ) \log_4\left(\frac{6}{y}\right)\ lo g 4 ( y 6 )

Expand log ⁡ 4 ( 3 x y ) \log_4\left(3xy\right) lo g 4 ( 3 x y )

3 log ⁡ 4 ( x ) + 3 log ⁡ 4 ( y ) 3\log_4\left(x\right)\ +\ 3\log_4\left(y\right) 3 lo g 4 ( x ) + 3 lo g 4 ( y )

Expand log ⁡ 2 ( x 3 y ) \log_2\left(\frac{x}{3y}\right) lo g 2 ( 3 y x )

3 log ⁡ 2 ( x ) + log ⁡ 2 ( y ) 3\log_2\left(x\right)\ +\ \log_2\left(y\right) 3 lo g 2 ( x ) + lo g 2 ( y )

log ⁡ 2 ( x ) − 3 log ⁡ 2 ( y ) \log_2\left(x\right)\ -\ 3\log_2\left(y\right) lo g 2 ( x ) − 3 lo g 2 ( y )

Use change of base property to write log ⁡ 16 log ⁡ 3 \frac{\log16}{\log3} lo g 3 lo g 1 6 as one Log

log ⁡ 3 ( 16 ) \log_3\left(16\right)\ lo g 3 ( 1 6 )

log ⁡ 16 ( 3 ) \log_{16}\left(3\right) lo g 1 6 ( 3 )

log ⁡ ( 13 ) \log\left(13\right) lo g ( 1 3 )

log ⁡ ( 19 ) \log\left(19\right) lo g ( 1 9 )

Use properties of Logs to write log ⁡ 3 ( 6 ) + log ⁡ 3 ( 2 ) \log_3\left(6\right)+\log_3\left(2\right) lo g 3 ( 6 ) + lo g 3 ( 2 ) as one Log

log ⁡ 3 ( 12 ) \log_3\left(12\right) lo g 3 ( 1 2 )

log ⁡ 3 ( 8 ) \log_3\left(8\right)\ lo g 3 ( 8 )

log ⁡ 3 ( 4 ) \log_3\left(4\right) lo g 3 ( 4 )

log ⁡ 3 ( 3 ) \log_3\left(3\right) lo g 3 ( 3 )

Use properties of Logs to write log ⁡ 3 ( 8 ) + log ⁡ 3 ( 3 ) − log ⁡ 3 ( 2 ) \log_3\left(8\right)+\log_3\left(3\right)-\log_3\left(2\right) lo g 3 ( 8 ) + lo g 3 ( 3 ) − lo g 3 ( 2 ) as one Log

log ⁡ 3 ( 12 ) \log_3\left(12\right)\ lo g 3 ( 1 2 )

log ⁡ 3 ( 9 ) \log_3\left(9\right) lo g 3 ( 9 )

log ⁡ 3 ( 48 ) \log_3\left(48\right) lo g 3 ( 4 8 )

log ⁡ 3 ( 8 ) \log_3\left(8\right) lo g 3 ( 8 )

Use properties of Logs to write log ⁡ 3 ( 4 ) + log ⁡ 3 ( 3 ) + log ⁡ 3 ( 2 ) \log_3\left(4\right)+\log_3\left(3\right)+\log_3\left(2\right) lo g 3 ( 4 ) + lo g 3 ( 3 ) + lo g 3 ( 2 ) as one Log

log ⁡ 3 ( 24 ) \log_3\left(24\right) lo g 3 ( 2 4 )

log ⁡ 3 ( 9 ) \log_3\left(9\right)\ lo g 3 ( 9 )

log ⁡ 3 ( 6 ) \log_3\left(6\right) lo g 3 ( 6 )

log ⁡ 3 ( 14 ) \log_3\left(14\right) lo g 3 ( 1 4 )

Use properties of Logs to write log ⁡ 3 ( 4 ) − log ⁡ 3 ( x ) + log ⁡ 3 ( y ) \log_3\left(4\right)-\log_3\left(x\right)+\log_3\left(y\right) lo g 3 ( 4 ) − lo g 3 ( x ) + lo g 3 ( y ) as one Log

log ⁡ 3 ( 4 y x ) \log_3\left(\frac{4y}{x}\right) lo g 3 ( x 4 y )

log ⁡ 3 ( 4 x y ) \log_3\left(\frac{4x}{y}\right)\ lo g 3 ( y 4 x )

log ⁡ 3 ( 4 x y ) \log_3\left(4xy\right) lo g 3 ( 4 x y )

log ⁡ 3 ( 4 x y ) \log_3\left(\frac{4}{xy}\right) lo g 3 ( x y 4 )

Use properties of Logs to write 2 log ⁡ 3 ( 4 ) 2\log_3\left(4\right) 2 lo g 3 ( 4 ) as one Log

log ⁡ 3 ( 16 ) \log_3\left(16\right) lo g 3 ( 1 6 )

log ⁡ 3 ( 12 ) \log_3\left(12\right)\ lo g 3 ( 1 2 )

log ⁡ 3 ( 8 ) \log_3\left(8\right) lo g 3 ( 8 )

log ⁡ 3 ( 6 ) \log_3\left(6\right) lo g 3 ( 6 )

Log Properties

Authored by Nicole Baker

Mathematics

10th - 12th Grade

CCSS covered

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MULTIPLE CHOICE QUESTION

2 mins • 1 pt

Use the property Log_b(xy) = Log_b(x) + Log_b(y)
to expand: Log₃(5x)

$\log_2\left(5\right)\ \times\ \log_2\left(x\right)$

$\log_2\left(5\right)\ +\ \log_2\left(x\right)$

$\log_2\left(5\right)\ -\ \log_2\left(x\right)$

$\log_2\left(5\ +\ x\right)\$

Tags

CCSS.HSA.SSE.B.3

CCSS.HSF.BF.B.5

CCSS.HSF.IF.C.8

CCSS.HSA.SSE.A.2

MULTIPLE CHOICE QUESTION

2 mins • 1 pt

Use the property $\log_b\left(\frac{x}{y}\right)\ =\ \log_b\left(x\right)\ -\ \log_b\left(y\right)$
to expand $\log_2\left(\frac{x}{4}\right)\$

$\log_2\left(x\right)\ \times\ \log_2\left(4\right)$

$\log_2\left(x\right)\ +\ \log_2\left(4\right)$

$\log_2\left(x\right)\ -\ \log_2\left(4\right)$

$\log_2\left(5\right)\ -\ \log_2\left(x\right)\$

Tags

CCSS.HSA.SSE.B.3

CCSS.HSF.BF.B.5

CCSS.HSF.IF.C.8

MULTIPLE CHOICE QUESTION

2 mins • 1 pt

Use the property $\log_b\left(x\right)^a\ =\ a\log_b\left(x\right)\$
to expand $\log_5\left(x\right)^2\$

$5\log_2\left(x\right)\$

$2\log_5\left(x\right)\$

$\log_5\left(x\right)\ +\ \log_5\left(2\right)$

$\log_5\left(x\right)\ -\ \log_5\left(2\right)\$

Tags

CCSS.HSA.SSE.B.3

CCSS.HSF.BF.B.5

CCSS.HSF.IF.C.8

CCSS.HSA.SSE.A.2

MULTIPLE CHOICE QUESTION

2 mins • 1 pt

Use the change of base property $\log_b\left(x\right)\ =\ \frac{\log_a\left(x\right)}{\log_a\left(b\right)}\$
to write $\log_2\left(16\right)\$ with the common base

$\log\left(8\right)$

$\frac{\log\left(16\right)}{\log\left(2\right)}$

$\frac{\log\left(2\right)}{\log\left(16\right)}$

$\log\left(16\right)\ +\ \log\left(2\right)\$

Tags

CCSS.HSF.BF.B.5

MULTIPLE CHOICE QUESTION

2 mins • 1 pt

Expand $\log_4\left(6y\right)$

$\log_4\left(6\right)\ +\ \log_4\left(y\right)$

$\log_4\left(6\right)\ -\ \log_4\left(y\right)$

$y\log_4\left(6\right)\$

$\log_4\left(\frac{6}{y}\right)\$

Tags

CCSS.HSA.SSE.B.3

CCSS.HSF.BF.B.5

CCSS.HSF.IF.C.8

CCSS.HSA.SSE.A.2

MULTIPLE CHOICE QUESTION

2 mins • 1 pt

Expand $\log_4\left(3xy\right)$

$\log_4\left(3\right)\ +\ \log_4\left(x\right)-\ \log_4\left(y\right)$

$3\log_4\left(x\right)\ +\ 3\log_4\left(y\right)$

$\log_4\left(3\right)\ +\ \log_4\left(x\right)+\ \log_4\left(y\right)\$

$4\log\left(3\right)\ +\ 4\log\left(x\right)+\ 4\log\left(y\right)\$

Tags

CCSS.HSA.SSE.B.3

CCSS.HSF.BF.B.5

CCSS.HSF.IF.C.8

MULTIPLE CHOICE QUESTION

2 mins • 1 pt

Expand $\log_2\left(3y^2\right)$

$\log_2\left(3\right)\ +\ \log_2\left(y\right)$

$2\log_2\left(3\right)\ +\ 2\log_2\left(y\right)$

$\log_2\left(3\right)\ +\ 2\log_2\left(y\right)$

$\log_2\left(\frac{3}{y^2}\right)\$

Tags

CCSS.HSA.SSE.B.3

CCSS.HSF.IF.C.8

CCSS.HSA.SSE.A.2

MULTIPLE CHOICE QUESTION

2 mins • 1 pt

Expand $\log_2\left(\frac{x}{3y}\right)$

$\log_2\left(x\right)\ +\ \log_2\left(3\right)\ +\ \log_2\left(y\right)$

$3\log_2\left(x\right)\ +\ \log_2\left(y\right)$

$\log_2\left(x\right)\ -\ 3\log_2\left(y\right)$

$\log_2\left(x\right)\ -\ \log_2\left(3\right)\ -\ \log_2\left(y\right)$

Tags

CCSS.HSA.SSE.B.3

CCSS.HSF.BF.B.5

CCSS.HSF.IF.C.8

CCSS.HSA.SSE.A.2

MULTIPLE CHOICE QUESTION

2 mins • 1 pt

Use change of base property to write $\frac{\log16}{\log3}$ as one Log

$\log_3\left(16\right)\$

$\log_{16}\left(3\right)$

$\log\left(13\right)$

$\log\left(19\right)$

Tags

CCSS.HSF.BF.B.5

10.

MULTIPLE CHOICE QUESTION

2 mins • 1 pt

Use properties of Logs to write $\log_3\left(6\right)+\log_3\left(2\right)$

as one Log

$\log_3\left(8\right)\$

$\log_3\left(4\right)$

$\log_3\left(12\right)$

$\log_3\left(3\right)$

Tags

CCSS.HSA.SSE.B.3

CCSS.HSF.BF.B.5

CCSS.HSF.IF.C.8

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Log Properties

Use the property Logb(xy) = Logb(x) + Logb(y)to expand: Log3(5x)

Use the property log⁡b(xy) = log⁡b(x) − log⁡b(y)\log_b\left(\frac{x}{y}\right)\ =\ \log_b\left(x\right)\ -\ \log_b\left(y\right)logb​(yx​) = logb​(x) − logb​(y) to expand log⁡2(x4) \log_2\left(\frac{x}{4}\right)\ log2​(4x​)

Use the property log⁡b(x)a = alog⁡b(x) \log_b\left(x\right)^a\ =\ a\log_b\left(x\right)\ logb​(x)a = alogb​(x) to expand log⁡5(x)2 \log_5\left(x\right)^2\ log5​(x)2

Use the change of base property log⁡b(x) = log⁡a(x)log⁡a(b) \log_b\left(x\right)\ =\ \frac{\log_a\left(x\right)}{\log_a\left(b\right)}\ logb​(x) = loga​(b)loga​(x)​ to write log⁡2(16) \log_2\left(16\right)\ log2​(16) with the common base

Expand log⁡4(6y)\log_4\left(6y\right)log4​(6y)

Expand log⁡4(3xy)\log_4\left(3xy\right)log4​(3xy)

Expand log⁡2(3y2)\log_2\left(3y^2\right)log2​(3y2)

Expand log⁡2(x3y)\log_2\left(\frac{x}{3y}\right)log2​(3yx​)

Use change of base property to write log⁡16log⁡3\frac{\log16}{\log3}log3log16​ as one Log

Use properties of Logs to write log⁡3(6)+log⁡3(2)\log_3\left(6\right)+\log_3\left(2\right)log3​(6)+log3​(2) as one Log

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Discover more resources for Mathematics

Use the property Log_b(xy) = Log_b(x) + Log_b(y)
to expand: Log₃(5x)

Use the property $\log_b\left(\frac{x}{y}\right)\ =\ \log_b\left(x\right)\ -\ \log_b\left(y\right)$
to expand $\log_2\left(\frac{x}{4}\right)\$

Use the property $\log_b\left(x\right)^a\ =\ a\log_b\left(x\right)\$
to expand $\log_5\left(x\right)^2\$

Use the change of base property $\log_b\left(x\right)\ =\ \frac{\log_a\left(x\right)}{\log_a\left(b\right)}\$
to write $\log_2\left(16\right)\$ with the common base

Expand $\log_4\left(6y\right)$

Expand $\log_4\left(3xy\right)$

Expand $\log_2\left(3y^2\right)$

Expand $\log_2\left(\frac{x}{3y}\right)$

Use change of base property to write $\frac{\log16}{\log3}$ as one Log

Use properties of Logs to write $\log_3\left(6\right)+\log_3\left(2\right)$

as one Log