# Logarithm and logarithm functions

This is a very important section so ensure that you learn it and understand it. We will begin by drawing up a curve for *y* = 10* ^{x}*.

We have also marked the point where y = 7 on the curve. By reading the *x*-axis we see that when *y* = 7 then *x* ≈ 0.85. Thus:

$$10^{0,85}\approx 7$$

We can take whatever (positive) value on the *y*-axis and read it off on the *x*-axis. We can take whatever number and rewrite it as exponential expression where 10 is the base, i.e. as an exponent of 10.

**Example**

$$10^{0}=1\; \;\;\;\;\, \;10^{0.78}\approx 6\\ 10^{0.3}\approx2\; \;\; \;10^{0.85}\approx 7\\ 10^{0.48}\approx3\; \;\;10^{0.9}\approx 8\\ 10^{0.6}\approx4\; \;\;\;10^{0.95}\approx 9\\ 10^{0.7}\approx 5\; \;\;\;10^{1}=10$$

The exponent is known as a *base 10 logarithm*. For example, in order to solve equations such as:

$$11=10^{x}$$

we must either solve it graphically, by drawing up a 10^{x} curve and find the value of *x* when *y* = 11 (as above); or we may use our pocket calculator that has a function which corresponds to manually drawing and reading of the graph. The key is designated as "lg" or "log". The solution to the equation is:

$$x=\log 11\approx 1.04$$

An equation of the form

$$y=log_{b}x$$

Is called a logarithmic function and when it is written as

$$y=log_{10}x$$

it is called base ten logarithm.

**Video lesson**

Base 10 logarithms