We want to solve
the given exponential equation
in terms of logarithms,
or correct to four decimal places.
So, our exponential equation
is 10 raised to the power of X
minus 10 equals six.
We want to begin by isolating
the exponential part,
meaning we want to isolate
10 raised to the power of X.
So, for the first step, we'll add 10
to both sides of the equation.
Simplifying, we now have seven times 10,
raised to the power of X equals 16.
Next, we'll divide both sides by seven
to isolate 10 raised to the power of X.
So, simplifying, we now have
10 raised to the power of X
equals 16/7.
And now, to solve for X,
we can take the common log
of both sides of the equation
because we have base 10 here,
or we can write this exponential equation
as the equivalent log equation
using the definition of a
logarithm, shown here below.
For this example, let's use
your definition of a logarithm
to solve for X.
Looking at our notes for a moment,
this exponential equation
in this log equation
are equivalent,
where b is the base, a is the exponent,
and n is the number.
So, let's write the
equivalent log equation.
We know we'll have a logarithm.
Because we have an equation,
we'll have an equals sign.
And now, we'll identify the
three parts of the equation.
We'll notice how the base is 10.
Log base 10 is called common
log, and so for the next step,
we can leave the base off because
if there is no base given,
we know it's common log, or log base 10.
A logarithm is an exponent,
so the logarithm is going to equal X.
Again, notice how a is the exponent,
and the logarithm is equal
to a, and then finally,
because the exponential is
equal to the number 16/7,
we have log base 10 of 16/7 equals X.
So notice how by writing
the equivalent log equation,
we now have an equation
that is solved for X,
and therefore, our exact
solution is X equals
common log of 16/7.
Again, common log is log
base 10, and for common log,
we don't need to write the base.
So this would be the exact solution for X.
Let's also get a decimal approximation
to four decimal places.
Here's the common log button,
so log 16 divided by seven,
close parenthesis, and enter.
To four decimal places, we
have approximately 0.3590.
Before we go, let's verify
that our solution is correct
by making the substitution
for X in the equation.
So, on the left side of the
equation we would have seven
and then times 10, raised to
the power of common log 16/7,
right arrow, then we have minus 10,
and this should equal positive six.
And notice how it does, verifying
our solution is correct.
I hope you found this helpful.
