We want to solve the exponential equation,
eight times e raised to the power of x
minus six equals 17.
The first step is to isolate the exponential part
of the equation, which means you want to isolate
e raised to the power of x.
So let's first add six to both sides of the equation.
Simplifying, we now have 8 e to the x equals 23.
Next we'll divide both sides by eight.
Simplifying again, we now have e to the x
equals 23 eighths.
Now for the next step, we can take the natural log
of both sides of the equation
or write the exponential equation as a log equation
using the definition of a logarithm shown here below.
Where the log equations and exponential equation
are equivalent where be is the base,
a is the exponent,
and n is the number.
So for this example let's write the equivalent log equation.
So the log equation will have a logarithm in it.
It's an equation so it will have an equal sign
and now let's identify the base, the exponent,
and the number.
Well the base is e, so we have log base e.
Log base e is the natural log,
so we'll use natural log in the next step.
A logarithm is an exponent.
The exponenet is x, so the logarithm is going to equal x.
And e to the x equals the number 23 eighths,
which means you have log base e of 23 eighths,
equals x.
And notice how now, the equation is solved for x.
To double check our work, if we were going to write
the log equation as an exponential equation,
we'd have base e raised to the power of x equals 23 eighths.
So our log equation is correct.
But let's us natural log notation
instead of using log base e.
Again log base e is natural log so we can write this as,
natural log of 23 eighths equals x.
then we want x equals natural log of 23 eighths.
Let's also get a decimal approximation for x.
To do this we'll use the calculator
and we'll enter natural log 23 divided by eight
closed parenthesis and enter.
To four decimal places, x is approximately 1.0561.
Let's also check our solution by verifying
this value of x satisfies the original equation.
So using the original equation
we'll substitute natural log 23 eighths for x.
So we'd have eight times e raised to the power
of natural log 23 eighths minus six equals 17.
Let's simplify e raised to the power
of natural log 23 eighths.
because the base is e and it's raised to the power
of natural log which is log base e,
this simplifies perfectly to 23 eighths.
So we have eight times 23 eighths minus six equals 17.
Multiplying here, as a fraction eight is eight over one.
Eight over eight simplifies to one.
So we have 23 minus 6 equals 17, which is true.
Let's also check this using the calculator
by entering the left side of the equation here
just to make sure it's equal to 17.
So we'd have eight 2nd natural log brings up,
e raised to the power of, then the exponent is
natural log 23 divided by eight closed parenthesis.
Right arrow minus six, enter.
And notice how we do get 17.
So the exact solution is natural log 23 eighths
or as a decimal approximation,
we would have approximately 1.0561.
And notice how the directions will take
either the exact solution or the approximate solution
to four decimal places.
I hope you found this helpful.