Solving Logarithmic and Exponential Equations

Front

What is the approximate value of \( \log 5 \)?

Back

Approximately 0.7.

Front

What is the approximate value of \( \log 8 \)?

Back

Approximately 0.9.

Front

How do you evaluate \( \log 200 \) using mental calculations?

Back

Use the approximations: \( \log 200 = \log(2 \times 100) = \log 2 + \log 100 = \log 2 + 2 \). With \( \log 2 \approx 0.3 \), you get \( \log 200 \approx 0.3 + 2 = 2.3 \).

Front

What is the result of condensing \( \log a + \log b \)?

Back

It can be condensed to \( \log(ab) \).

Front

How do you evaluate \( \log \frac{64}{5} \) using mental calculations?

Back

Use the approximations: \( \log \frac{64}{5} = \log 64 - \log 5 = 1.2 - 0.7 = 1.1 \).

Front

What is the solution to the equation \( \log x + \log 8 = 2 \)?

Back

Combine the logs: \( \log(8x) = 2 \) implies \( 8x = 100 \) or \( x = 12.5 \).

Front

How do you expand the logarithm \( \log_{10} 7x \)?

Back

It can be expanded to \( \log_{10} 7 + \log_{10} x \).

Front

What is the change of base formula for logarithms?

Back

The change of base formula states that \( \log_b a = \frac{\log_k a}{\log_k b} \) for any positive base \( k \).

Front

What is the relationship between exponential and logarithmic functions?

Back

If \( y = b^x \), then \( x = \log_b y \). They are inverse functions.

Front

What is the value of \( \log_{10} 1000 \)?

Back

The value is 3, since \( 10^3 = 1000 \).