Introduction to Logarithms: Practice

The base stays the base, and the exponent becomes the value the logarithm equals.

$2^{4} = 16 \iff \log_{2}(16) = 4$

The base is $3$, the logarithm equals the exponent $2$, and the input $9$ is the result.

$\log_{3}(9) = 2 \iff 3^{2} = 9$

In $\log_{b}(y)$ the small number written below is the base $b$.

$\log_{4}(64) \;\Rightarrow\; \text{base} = 4$

The number $3$ is what the log equals, since $4^{3} = 64$.

The natural log has base $e \approx 2.718$.

$\ln(x) = \log_{e}(x)$

Ask: three to what power is $27$?

$3^{3} = 27 \;\Rightarrow\; \log_{3}(27) = 3$

You can take the log only of a positive number, and $0$ is not positive.

$\log_{b}(0) \text{ is undefined}$

The small value $0.1$ is fine; its log is simply negative.

The log of $1$ is $0$ in every base, because any base to the zero power is $1$.

$e^{0} = 1 \;\Rightarrow\; \ln(1) = 0$

A base to the first power is itself.

$6^{1} = 6 \;\Rightarrow\; \log_{6}(6) = 1$

Ask: two to what power is $16$?

$2^{4} = 16 \;\Rightarrow\; \log_{2}(16) = 4$

The base is $5$, the exponent is $2$, and the result is $25$.

$\log_{5}(25) = 2 \iff 5^{2} = 25$

The definition of a logarithm links the two forms directly.

$\log_{b}(y) = x \iff b^{x} = y$

Ask: four to what power is $16$?

$4^{2} = 16 \;\Rightarrow\; \log_{4}(16) = 2$