The laws of logarithms

A sum of logarithms to the same base is the logarithm of the product:

• The law comes from the law of powers $\displaystyle {a^m\cdot a^n}={a^{m+n}}$: if $\displaystyle {\log_a x}={m}$ and $\displaystyle {\log_a y}={n}$, then $\displaystyle {x}={a^m}$, $\displaystyle {y}={a^n}$, and $\displaystyle {xy}={a^{m+n}}$. So $\displaystyle {\log_a(xy)}={m+n}$.
• $\displaystyle {\log_6 4+\log_6 9}={\log_6 36}={2}$: neither logarithm is a whole number, and their sum is 2.

A difference of logarithms to the same base is the logarithm of the quotient:

• The law comes from $\displaystyle {\frac{a^m}{a^n}}={a^{m-n}}$.
• $\displaystyle {\log_2 24-\log_2 3}={\log_2 8}={3}$.

The number’s exponent comes out in front of the logarithm:

• The law comes from $\displaystyle {(a^m)^b}={a^{mb}}$.
• $\displaystyle {2\log5+\log4}={\log5^2+\log4}={\log25+\log4}={\log100}={2}$.

• $\displaystyle \log(x+y)$ is not $\displaystyle \log x+\log y$: $\displaystyle {\log(10+10)}={\log20\approx1.30}$, while $\displaystyle {\log10+\log10}={2}$.
• $\displaystyle \log x\cdot\log y$ is not $\displaystyle \log(xy)$.
• $\displaystyle \frac{\log x}{\log y}$ is not $\displaystyle \log\frac{x}{y}$.
• The laws hold only where every logarithm in them is defined.

The laws work both ways: from several logarithms to one, and from one logarithm to a sum.

• $\displaystyle {\log_2(8x)}={\log_2 8+\log_2 x}={3+\log_2 x}$ (where $\displaystyle x>0$).
• $\displaystyle {\log_3(9x^2)}={\log_3 9+2\log_3 x}={2+2\log_3 x}$ (where $\displaystyle x>0$).
• $\displaystyle {\log x+\log(x-3)}={\log(x(x-3))}={\log(x^2-3x)}$ (where $\displaystyle x>3$).

In $\displaystyle {3^x}={20}$, 20 cannot be written as a whole power of 3. The solution below takes $\displaystyle \log$ of both sides, and brings $\displaystyle x$ down from the exponent by the law of the logarithm of a power:

• $\displaystyle {x\cdot\log3}={\log20}$.
• Divide by $\displaystyle \log3$: $\displaystyle {x}={\frac{\log20}{\log3}}$ – the exact answer.
• With a calculator: $\displaystyle x\approx2.73$. A check: $\displaystyle 3^{2.73}\approx20$.

Calculate with the laws of logarithms.

Write as one logarithm, or as a number times one logarithm ($\displaystyle x>0$).

Write $\displaystyle \log_2(8x)$ as the sum of a number and a logarithm ($\displaystyle x>0$).

The equation is $\displaystyle {5^x}={8}$.

Solve it by the logarithm of both sides, and write the exact answer.

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

Solve the equation $\displaystyle {7^x}={2\cdot3^x}$.
Write the exact answer.