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Section 6.3 : Solving Exponential Equations

5. Solve the following equation.

\[{2^{3x}} = 10\]

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Start Solution

For this equation there is no way to easily get both sides with the same base. Therefore, we’ll need to take the logarithm of both sides.

We can use any logarithm and the natural logarithm and common logarithm are usually good choices since most calculators can handle them. Because one of the bases in this equation is a 10 the common logarithm will probably be the better choice (although we can use the natural logarithm if we wanted to).

Taking the logarithm (using the common logarithm) of both sides gives,

\[\log {2^{3x}} = \log 10\] Show Step 2

Now we can easily compute the right side (which is also why we chose the common logarithm for this case) and we can use the logarithm property that says,

\[{\log _b}{x^r} = r{\log _b}x\]

to move the 3\(x\) out of the exponent from the logarithm on the left. Doing this gives,

\[3x\left( {\log 2} \right) = x\left( {3\log 2} \right) = 1\]

We did a little rearranging of the left side to put all the numbers together in order to make the next step a little easier.

Show Step 3

Finally, all we need to do is solve for \(x\). Recall that the equations at this step tend to look messier than we are used to dealing with. However, the logarithms in the equation at this point are just numbers and so we treat them as we treat all numbers with these kinds of equations.

In other words, all we need to do is divide both sides by the coefficient of the \(x\) and then user our calculators to get a decimal answer.

Here is the rest of the work for this problem.

\[x\left( {3\log 2} \right) = 1\hspace{0.25in} \to \hspace{0.25in}x = \frac{1}{{3\log 2}} = \frac{1}{{3\left( {0.301029996} \right)}} = \require{bbox} \bbox[2pt,border:1px solid black]{{1.10730936}}\]