Reduction to Linear Form

Some exponential equations can be reduced to a form that looks like $y=mx+c$ . Specifically, after applying the laws of logarithms we can treat $y=ax^{n}$ and $y=ab^{x}$ as if they were linear equations.

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$mathbf{y=ax^{n}}$

$y=ax^{n}$

Take logarithms:

$begin{aligned}log(y)&=log(ax^{n})[1.2em]&=log(a)+log(x^{n})[1.2em]&=log(a)+nlog(x)end{aligned}$

Overall we have:

$y=ax^{n}Rightarrow log(y)=log(a)+nlog(x)$

Now if we plot $log(y)$ against $log(x)$ , we have a straight line graph.

The graph below shows $y$ against $x$ in red and $log(y)$ against $log(x)$ in blue. As expected, the blue line is straight.

$mathbf{y=ab^{x}}$

$y=ab^{x}$

Take logarithms:

$begin{aligned}log(y)&=log(ab^{x})[1.2em]&=log(a)+log(b^{x})[1.2em]&=log(a)+xlog(b)end{aligned}$

Overall we have:

$y=ab^{x}Rightarrow log(y)=log(a)+xlog(b)$

Now if we plot $log(y)$ against $x$ , we have a straight line graph.

The graph below shows $y$ against $x$ in red and $log(y)$ against $x$ in blue. As expected, the blue line is straight.

Example 1: Converting to Linear Form

Convert $y=99times0.5^{x}$ to linear form.

[2 marks]

$y=99times0.5^{x}$

$begin{aligned}log(y)&=log(99times0.5^{x})[1.2em]&=log(99)+log(0.5^{x})[1.2em]&=log(99)+xlog(0.5)end{aligned}$

Example 2: Using Linear Form

The data below is taken from a plot of the form $y=ax^{n}$ . By plotting $log_{10}(y)$ against $log_{10}(x)$ , find $a$ and $n$ .

[5 marks]

Step 1: Calculate $log_{10}(x)$ and $log_{10}y$ and put them in a table.

Step 2: Plot $log(y)$ against $log(x)$ on a scatter graph.

Step 3: Use the scatter graph to find the gradient and y-intercept.

Gradient is approximately $dfrac{3.3-0.6}{0.9-0}=dfrac{2.7}{0.9}=3$

y-intercept is approximately $0.6$

Step 4: Use our linear form to interpret the gradient and y-intercept.

$y=ax^{n}rightarrow log(y)=log(a)+nlog(x)$

Gradient is $n$ so $n=3$

y-intercept is $log(a)$ so $log(a)=0.6$ so $a=10^{0.6}=4$ to two significant figures.

Step 5: Put together to determine the form of the plot:

$y=4x^{3}$

Reduction to Linear Form Example Questions

Question 1: The number of branches of a high-street store decreases over time. This trend, which is of the form $y=ab^{x}$ is monitored over a number of years. Use the data from the monitoring to find $a$ and $b$ .

[5 marks]

If the question does not specify a base for logarithms it is up to us to choose a sensible base. The choice of base should not affect the answer at the end. In the working below, we have used a base of $2$ .

Note the linear form:

$y=ab^{x}rightarrow log(y)=xlog(b)+log(a)$

So the straight line is obtained by plotting $log(y)$ against $x$ .

This means that we need to add a $log(y)$ row to the table.

Plot $log(y)$ against $x$ and add a line of best fit.

Find the gradient and the y-intercept.

Gradient is $dfrac{6-9.5}{6-0}=dfrac{3.5}{6}=-dfrac{7}{12}$

y-intercept is $9.5$

Interpret the gradient and y-intercept.

Gradient is $log(b)$

$log(b)=-dfrac{7}{12}$

$b=0.667$

y-intercept is $log(a)$

$log(a)=9.5$

$a=729$

Put it all together:

Our estimate for the line is $y=729times0.667^{x}$

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Question 2: The value of an investment in MathCoin is believed to reliably climb following an $ax^{n}$ trajectory. Claire buys one MathCoin and monitors the price of her investment every month over one year. Find $a$ and $n$ .

[5 marks]

If the question does not specify a base for logarithms it is up to us to choose a sensible base. The choice of base should not affect the answer at the end. In the working below, we have used a base of $10$ .

Note the linear form:

$y=ax^{n}rightarrow log(y)=nlog(x)+log(a)$

So the straight line is obtained by plotting $log(y)$ against $log(x)$ .

This means that we need to add a $log(x)$ row and a $log(y)$ row to the table.

Plot $log(y)$ against $log(x)$ and add a line of best fit.

Find the gradient and the y-intercept.

Gradient is $dfrac{3.5-2}{1-0}=dfrac{1.5}{1}=1.5$

y-intercept is $2$

Interpret the gradient and y-intercept.

Gradient is $n$

$n=1.5$

y-intercept is $log(a)$

$log(a)=2$

$a=100$

Put it all together:

Our estimate for the line is $y=100times x^{1.5}$

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Question 3: A forest has been declining in population since the nineteenth century. To assess this decline, a census of the tree population of the forest was conducted every ten years in the twentieth century. It is believed this decline follows a $y=ab^{x}$ curve. Use the table of census data to find $a$ and $b$ .