The logarithm

In the equation $\displaystyle {2^x}={8}$ the base, 2, and the result, 8, are known; the exponent is missing. It is 3, since $\displaystyle {2^3}={8}$.

• This exponent is called the logarithm of 8 to the base 2, written $\displaystyle {\log_2 8}={3}$.
• Taking a logarithm is the inverse of raising to a power: a power gives the result from the base and the exponent, a logarithm gives the exponent from the base and the result.

$\displaystyle \log_a x$ is the exponent to which $\displaystyle a$ must be raised to give $\displaystyle x$:

• $\displaystyle a$ is called the base of the logarithm.
• The base is positive and not 1, as the base of an exponential function is: every power of 1 is 1, so $\displaystyle \log_1 5$ would have no value.

To calculate a logarithm we write the number as a power of the base:

• $\displaystyle {\log_2 32}={5}$, since $\displaystyle {2^5}={32}$.
• $\displaystyle {\log_3\frac{1}{9}}={-2}$, since $\displaystyle {3^{-2}}={\frac{1}{9}}$.
• $\displaystyle {\log_9 27}={\frac{3}{2}}$, since $\displaystyle {9^{\frac{3}{2}}}={\left(\sqrt{9}\right)^3}={27}$.
• $\displaystyle {\log_5 1}={0}$, since $\displaystyle {5^0}={1}$; the same in every base: $\displaystyle {\log_a 1}={0}$.
• $\displaystyle {\log_7 7}={1}$, since $\displaystyle {7^1}={7}$; the same in every base: $\displaystyle {\log_a a}={1}$.

A power of a positive base is always a positive number: $\displaystyle a^b>0$ for every $\displaystyle b$.

• So no exponent $\displaystyle b$ gives $\displaystyle {a^b}={0}$ or $\displaystyle {a^b}={-4}$.
• $\displaystyle \log_a x$ is defined only when $\displaystyle x>0$: $\displaystyle \log_2 0$ and $\displaystyle \log_2(-4)$ have no value.
• The logarithm itself may be negative: $\displaystyle {\log_2\frac{1}{8}}={-3}$.

A logarithm to the base 10 is written without its base: $\displaystyle \log x$ means $\displaystyle \log_{10}x$.

• $\displaystyle {\log1000}={3}$, $\displaystyle {\log0.01}={-2}$.
• A calculator’s log key gives the logarithm to the base 10 of any positive number: $\displaystyle \log50\approx1.699$.
• $\displaystyle \log50$ is between 1 and 2, since 50 is between $\displaystyle 10^1$ and $\displaystyle 10^2$.

The logarithm is the exponent, so the two operations, one after the other, give back the number we started from:

• $\displaystyle {5^{\log_5 7}}={7}$.
• $\displaystyle {\log_3 3^8}={8}$.

An equation whose unknown is inside a logarithm, or in an exponent, is solved by the definition: we go from the logarithm form to the power form, or back.

• $\displaystyle {\log_2 x}={5}$: by the definition $\displaystyle {x}={2^5}$, that is $\displaystyle {x}={32}$.
• $\displaystyle {\log_x 49}={2}$: by the definition $\displaystyle {x^2}={49}$. A base is positive, so $\displaystyle {x}={7}$ ($\displaystyle {x}={-7}$ is no base).
• $\displaystyle {2^x}={7}$: the exponent to which 2 is raised to give 7 is, by the definition, $\displaystyle {x}={\log_2 7}$. This is the exact answer. Since $\displaystyle {2^2}={4}$ and $\displaystyle {2^3}={8}$, its value is between 2 and 3.

Calculate.

Solve the equations.

The equation is $\displaystyle {10^x}={50}$.

Solve it, and write the exact answer.

Calculate $\displaystyle x$ with a calculator, to two decimal places.

Between which two consecutive whole numbers is $\displaystyle \log_2 20$? Explain without a calculator.