Change of Base Law

The log rules you have seen so far assume that each log has the same base, in some scenarios this will not be the case.

The Change of Base Law allows us to rewrite a logarithm with one base into terms of logarithms with a different base. This is particularly useful when solving logarithmic problems on a calculator, as many calculators only support logarithms with base 10 ($log$) or base e ($ln$).

The formula is:

$$\log_bx=\frac{\log_ax}{\log_ab}$$

Where:

• $b$ is the original base,
• $a$ is the new base,
• $x$ is the argument of the logarithm.

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Why it Works:

The change of base law comes from the definition of logarithms. Since logarithms are exponents, we can express:

$x = b^k \implies \log_b(x) = k$

By rewriting $b^k$ in terms of another base $a$, we get:

$x = b^k = (a^{\log_a(b)})^k \implies \log_b(x) = \frac{\log_a(x)}{\log_a(b)}$

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How to Use It:

1. Choose a base $a$ for convenience. Typically, a=10 (common log) or a=e (natural log).
2. Apply the formula $\log_b(x) = \frac{\log_a(x)}{\log_a(b)}$
3. Use a calculator to compute the values of $log_a(x)$ and $log_a(b)$.

Example