Some Techniques for Solving Exponential and Logarithmic Equations
by Shelley Walsh ©2000
For the following example problems the instruction is to solve for x algebraically.
Example 1:
Solution:
Explanation:
Here all of the numbers in the problem are powers of 3, so even though log base 3 is not on the calculator we can solve the problem by taking logs base 3 on both sides, because we can find the logs base 3 of all of the numbers involved by thinking about it. The left side of line 2 comes from one of the inverse properties of logarithms. I have colored the expression in the exponent to make it easier to see as representing a single number, because sometimes it is difficult to see that anything you can do with x you can do with 2x-1. For the rest of the solution we just solve it like any other linear equation, adding 1 to both sides and then dividing both sides by 2.Example 2:
Solution:
Explanation:
The numbers in this problem are not as nice as those in Example 1, because there is no number that 5 and 13 are nice powers of. In this kind of problem it is best to take log on both sides with a base that the calculator can do, either the common log or the natural log. I have chosen here to use the common log, the log base 10, but it would have worked just as well to have use the natural log, the log base e. Then instead of using an inverse property of logs we use the property about logs of powers. Again I have colored the 2x in hopes that you can more easily see it as behaving just like a single variable. Anything you can do with x you can do with 2x, because if x is a real number 2x is a real number too. In the 3rd line I rearranged things to make the 2log5 look more like a proper coefficient. Then in the next step we divide both sides by it and then get out the calculator to get a 4 place decimal approximation to the answer.Example 3:
Solution:
Explanation:
With e in the problem it is best to use the natural log ln, since ln means log base e. First we divide both sides by 6 to get the exponential alone. In the last line I used a little trick in order to save calculator keystrokes. It was not entirely necessary. It would also be fine to just key in .5 in the calculator and then write down the opposite of what you get.Example 4:
Solution:
Explanation:
With a logarithmic equation usually the best way to solve it is to exponentiate both sides, that is turn both sides into exponents for the base of the logarithm, in this case 3. Note that this is not at all the same thing as cubing both sides. Then we get the left side of the second line from one of the inverse properties and the right side is simply raising 3 to the second power, that is multiplying it by itself.Example 5:
Solution:
Explanation:
In this problem, before we can get rid of the logs we need to use the log properties to write the left side a single logarithm. It wouldn't do to just raise 2 to each term. You can only do that with multiplying both sides of an equation by something because of the distributive property, and there is no distributive power of exponentiating. So what we have to do is use the product property backwards to get the left side of our first line. After that we can exponentiate both sides with a base of 2 to get rid of the log base 2. After that it becomes a quadratic equation that we can solve by factoring. But the -2 solution has to be thrown out, because in the original equation it would require taking a log of a negative number, which is not allowed.Example 6:
Solution:
