Exercise 9 Page 480

First we will describe the error made, then we will solve the equation in a correct way.

Let's analyze the given solution step by step, and look for the error that was made.

ln 4 x = 5 \Leftrightarrow e^{l n 4 x} = e^{5}

The first step is correct. The natural base

e

was raised to the power of both sides of the first equation. Let's take a look at the next step.

e^{l n 4 x} = e^{5} \Leftrightarrow 4 x = 5

This is the step in which the error was made. The left-hand side of the equation was simplified correctly —

e

raised to the power of

ln a

is equal to the argument of the natural logarithm,

a

. However, there are no properties that would allow us to simplify the right-hand side of the equation as it was done.

Now we will solve the given equation.

ln 4 x = 5

Our first step will be the same as the first step in the given solution.

ln 4 x = 5 \Leftrightarrow e^{l n 4 x} = e^{5}

Next, we will simplify the left-hand side of the equation. Recall that

e

raised to the power of

ln a

is equal to the argument of the natural logarithm,

a

.

e^{l n 4 x} = e^{5} \Leftrightarrow 4 x = e^{5}

Finally, let's isolate

x

on the left-hand side of the equation.

4 x = e^{5}

$LHS / 4 = RHS / 4$

x = \frac{e ^{5}}{4}

Use a calculator

x = 37.10328 \dots

Round to $1$ decimal place(s)

x \approx 37.1

The solution to the given equation is

x \approx 37.1 .