Introduction to Logarithms: Practice

Ask: nine to what power is $3$? A square root is the power $\tfrac{1}{2}$.

$9^{1/2} = \sqrt{9} = 3 \;\Rightarrow\; \log_{9}(3) = \tfrac{1}{2}$

The logarithm reports the exponent on its own base, so it undoes the power directly.

$\log_{7}(7^{5}) = 5$

Raising the base to the logarithm of a number returns that number.

$10^{\log(4)} = 4$

The quotient law turns a division inside the log into a subtraction of logs.

$\log_{b}\left(\tfrac{M}{N}\right) = \log_{b}(M) - \log_{b}(N)$

The power law moves the exponent to the front as a coefficient.

$\log_{b}(x^{5}) = 5\log_{b}(x)$

The product law combines the sum into a single log of a product.

$\log_{2}(4) + \log_{2}(8) = \log_{2}(32) = 5$

The last step uses $2^{5} = 32$.

Write $32$ as a power of $2$ and match the exponents.

$2^{x} = 32 = 2^{5} \;\Rightarrow\; x = 5$

A reciprocal needs a negative exponent.

$3^{-2} = \frac{1}{9} \;\Rightarrow\; \log_{3}\left(\tfrac{1}{9}\right) = -2$

Write $1000$ as a power of ten and match the exponents.

$10^{x} = 1000 = 10^{3} \;\Rightarrow\; x = 3$

The log of the base raised to a power returns that power.

$\log_{b}(b^{9}) = 9$

As the input shrinks toward zero, the logarithm dives down without bound, hugging the vertical axis.

$x = 0$

This mirrors the exponential's horizontal asymptote $y = 0$.

The common log has base $10$; ask ten to what power is $10000$.

$10^{4} = 10000 \;\Rightarrow\; \log(10000) = 4$