Logarithms and exponentials

Logarithmic and exponential functions

Logarithms and indices

A logarithm is a power. These two statements say the same thing:

So $\log_2 8 = 3$ because $2^3 = 8$. Two values follow from the definition: $\log_a a = 1$ and $\log_a 1 = 0$.

The laws of logarithms

$\log_3[x(x - 2)] = 1 \quad\Rightarrow\quad x(x - 2) = 3 \quad\Rightarrow\quad (x - 3)(x + 1) = 0$

$x = -1$ would need the log of a negative number, so $x = 3$ only.

$\mathrm{e}^x$ and $\ln x$

The number $\mathrm{e} \approx 2.718$ is the base for which $y = \mathrm{e}^x$ has gradient equal to $y$ at every point. The natural logarithm $\ln x$ means $\log_\mathrm{e} x$.

$2x = \ln 7$, so $x = 0.973$. For the second, $3x - 1 = \mathrm{e}^2$, so $x = \dfrac{\mathrm{e}^2 + 1}{3} = 2.80$.

The unknown in a power

Take logarithms of both sides to bring the power down.

$(2x + 1)\ln 3 = x\ln 5 \quad\Rightarrow\quad x(2\ln 3 - \ln 5) = -\ln 3 \quad\Rightarrow\quad x = \frac{-\ln 3}{2\ln 3 - \ln 5} = -1.87$

Some equations are quadratic in a power.

Let $u = 3^x$: $u^2 - 4u + 3 = 0$, so $u = 1$ or $u = 3$. Then $3^x = 1$ gives $x = 0$, and $3^x = 3$ gives $x = 1$.

Inequalities with the unknown in a power

The working is the same, but dividing by a negative logarithm reverses the inequality.

$n\ln 0.8 < \ln 0.1 \quad\Rightarrow\quad n > \frac{\ln 0.1}{\ln 0.8} = 10.3$

The sign reversed because $\ln 0.8$ is negative. So $n = 11$.

Turning a relationship into a straight line

Taking logarithms turns a curve into a straight line of the form $Y = mX + c$. The constants come from the gradient and intercept.

The gradient is $n = \dfrac{7 - 3}{3 - 1} = 2$. The line is $\ln y = 2\ln x + 1$, so $\ln a = 1$ and $a = \mathrm{e} = 2.72$.