Log graphs, in various formats, are part of the new Edexcel syllabus.

Recall that e=2.718 to 3 decimal places and logs can be thought of as powers. See more on e or more on logs.

Hence, we can sketch the graphs of $y=e^x$ and $y=\ln(x)$ where the natural log, $\ln(x)$, is the log that has base e. Subsequently, we can also write $ln(x)$ as $\log_e(x)$.

$\ln(x)$ is the inverse of $e^x$ and so these graphs are reflections of each other in the line y=x.

Since $\ln(x)$ and $e^x$ are mathematical inverses we have that

$\ln\left(e^x\right)=e^{\ln(x)}=x$

## Example 1 – Other Log Graphs

Consider the equation $y=ax^n$.

In the same way that you can plot y against x, it is possible to plot log(y) against log(x). Note that the bases are missing as it works for any base (provided the same base is used for both log(x) and log(y)). To plot log(y) against log(x), start by logging both sides of the equation $y=ax^n$:

$\log(y)=\log(ax^n)$
$\Longrightarrow \log(y)=\log(a)+\log(x^n)$
$\Longrightarrow \log(y)=\log(a)+n\log(x)$

Recall that when plotting y against x, in the equation y=mx+c, m is the gradient and c is the y-intercept. In addition, we can write log(y) as nlog(x)+log(a) and so, in the plot of log(y) against log(x), n is the gradient and log(a) is the y-intercept.
logline1

Note that $ax^n\ne(ax)^n$ so when using the log rules above, multiplication goes first.

## Example 2 – Other Log Graphs

Consider the equation $y=kb^x$.

In the same way that you can plot y against x, it is possible to plot log(y) against x, in a similar way to Example 1, by taking the log (of any base) of both sides of the equation $y=kb^x$:

$\log(y)=\log(kb^x)$
$\Longrightarrow \log(y)=\log(k)+\log(b^x)$
$\Longrightarrow \log(y)=\log(k)+x\log(b)$

Recall that when plotting y against x, in the equation y=mx+c, m is the gradient and c is the y-intercept. In addition, log(y) can be written as log(b)x+log(k) and so, in the plot of log(y) against x, log(b) is the gradient and log(k) is the y-intercept.

logline2

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