 Hong Kong
Stage 4 - Stage 5

# Solving Log and Exponential Equations with Technology

Lesson

When using technology to solve any type of equation, we can either use a graphical approach, or we can use the numerical solving capabilities of our CAS calculators. Solving equations involving either exponential functions or logarithmic functions is no different.

## Solving an equation graphically

##### Example

Solve the equation $2e^{0.5x}=4^x$2e0.5x=4x graphically.

We first recognise that this equation represents two different exponential functions that have been equated to determine at which $x$x value(s) their $y$y values will be equal. In other words, graphically we're interested in where these two functions intersect.

We therefore enter each side of the equation as a different function in the graphing facility of our calculator. We now adjust our axes to get a nice view of our two graphs, and in particular where they intersect. To find where they intersect, we use the relevant commands on our CAS calculator. In this case, we find that the functions intersect where $x=0.7821$x=0.7821.

Note, we do not state the $y$y values. When two functions are equated to each other, this means that we're only interested in for what $x$x value(s) the functions have the same $y$y values.

## Solving an equation using Numerical Solver

Most of the time when solving an equation and you're given the option to solve it however you'd like, you'll choose to use the Numerical Solver on your CAS calculator. It's much faster and more convenient. Let's take a look at a quick example.

##### Example

Solve $\ln\left(x-1\right)=e^{-2x}$ln(x1)=e2x

On the ClassPad, there's groups of ways to input this in to the solving capability of the calculator. One methods is illustrated below. Then we simply input our equation, check we're solving for the correct variable, press solve and the calculator use a numerical solving algorithm to find the answer for you.  So our solution is $x=2.0178$x=2.0178

#### Worked Examples

##### question 1

Use a graphing calculator to find the solution(s) of $3x+2=4^x$3x+2=4x.

Give the solution(s) correct to the nearest tenth. If there is more than one solution, write all of them on the same line, separated by commas.

##### question 2

Use a graphing calculator to find the solution(s) of $\log x=x^2-10x+7$logx=x210x+7.

Give the solution(s) correct to the nearest hundredth. If there is more than one solution, write all of them on the same line, separated by commas.