12 Logarithms
Logarithms, or logs for short, are the same as powers just written in another way.
12.1 Reverse of indices
If \(a^y =x\) then \(y = \log_{a} x\).
\(a\) is called the base of the logarithm. When dealing with logs it’s often useful to think of a numerical example to keep the idea straight in your head.
\[ \begin{aligned} 10^3 &= 1000 \\ 3 &= \log_{10} 1000 \end{aligned} \] This is the same fact written in index notation and as a logarithm.
12.2 Rules of logarithms
Just as there are rules when dealing with indices, there are the corresponding rules when dealing with logarithms too.
- \(\log_{a} x + \log_{a} y = \log_{a} xy\)
- \(\log_{a} x - \log_{a} y = \log_{a}{\frac{x}{y}}\)
- \(\log_{a} x^n = n\log_{a} x\)
We can use these rules to manipulate algebraic expressions. For example, let’s write the following as a single logarithm:
\[ \begin{aligned} 3\log_{10} 2 + \log_{10} 5 - \log_{10} 4 &= \log_{10} 2^3 + \log_{10} 5 - \log_{10} 4 \\ &= \log_{10} 8 + \log_{10} 5 - \log_{10} 4 \\ &= \log_{10} (8 \times 5) - \log_{10} 4 \\ &= \log_{10} 40 - \log_{10} 4 \\ &= \log_{10} (\frac{40}{4}) \\ &= \log_{10} (10) \\ &= 1 \end{aligned} \] This is how it was done:
- First we used the power rule \(\log_{a} x^n = n\log_{a} x\),
- then the addition rule \(\log_{a} x + \log_{a} y = \log_{a} xy\),
- and finally, the subtraction rule \(\log_{a} x - \log_{a} y = \log_{a}{\frac{x}{y}}\).
- Then notice \(\log_{10} (10)= 1\) since \(10^1=10\).
Have a go at these simplification questions.
12.3 Solving equations with logarithms in
For example, let’s solve \(3\log_{10} x + \log_{10} 2 = \log_{10} 250\). First we’ll apply the power rule \(\log_{a} x^n = n\log_{a} x\), then the addition rule \(\log_{a} x + \log_{a} y = \log_{a} xy\):
\[ \begin{aligned} 3\log_{10} x + \log_{10} 2 &= \log_{10} 250 \\ \log_{10} x^3 + \log_{10} 2 &= \log_{10} 250 \\ \log_{10} 2x^3 &= \log_{10} 250 \end{aligned} \]
Now since the two sides are equal the values inside the logarithm must be equal. We can then go ahead and solve the resulting equation as normal.
\[ \begin{aligned} \log_{10} 2x^3 &= \log_{10} 250 \\ 2x^3 &= 250 \\ x^3 &= 125 \\ x &= \sqrt[3]{125} \\ &= 5 \end{aligned} \]
Have a go at the following questions:
12.4 Some important bases
Some bases in logarithms come up more than others, because of that some bases have their own notation.
12.4.1 The natural logarithm
A logarithm that has \(e\) as it’s base is known as the natural logarithm and has it’s own symbol.
\[ \log_{e} x = \ln{x} \]
12.4.2 Base 10
A logarithm that has \(10\) as it’s base has it’s own symbol.
\[ \log_{10} x = \log{x} \]
You just don’t bother writing the base.
12.5 Differentiating \(\ln{x}\)
The rule for differentiating \(\ln{x}\) is:
if \(y = k\ln{ax}\) then \(\frac{dy}{dx}= \frac{k}{x}\).
Use that rule to try the following questions.