# How to Eliminate Exponents in Calculus

Calculus > Eliminate Exponents in Calculus

Exponents can be a tricky factor in dealing with equations, and when exponents have variables in them it becomes even more complicated. It’s possible to eliminate some exponents using the The Power Rule, but this won’t work for exponents over 2. Another way to eliminate exponents in calculus is to convert exponential variables into a more manageable form, known as the logarithm function.

Graph of common log, by Ellywa|nl|Wikimedia Commons

A logarithm represents the value y the base, b, has to be raised to to equal the value x in the form of log_b(x). If x=by, then log_b(x)=y. Not every log is listed with a base b, and these are considered to use the common base of 10. The log might also appear in the form ln(x), which is a log taken to the base of e, the natural number.

## How to Eliminate Exponents in Calculus: Example

Sample Problem: Solve for the value of x if 10 to the 5x power plus 10 is equal to 20.

Step 1: Set up the equation from the information given in the question.
105x + 10 = 20

Step 2: Take 10 from both sides to eliminate the 10 near the variable. This is a basic algebra step, but still an important one.
105x + 10 – 10 = 20 – 10
giving:
105x = 10

Step 3: Take the log of both sides.
log(105x) = log(10)

Step 4: Apply the rule that states log_b(ac) = c * log_b(a). Using this, we can move the variable out of the exponent and leave it in a form we can simplify. If you recall that a log without a subscript is considered a base of 10, you can easily simplify log_10(10) = y as 1, due to by = x being 101 = 10.
5x * 1 = 1

Step 5: Divide both sides by 5 to isolate the variable. This will give you a final answer of 1/5, or .2.
5x/5 = 1/5 -> x = 1/5 = 0.2

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