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Solving equations like this usually starts with transforming the equation into one of the following general forms:
log(expression) = number
or
log(expression) = log(other-expression)
Since all our terms are logarithms we will aim for the "all-log" second form.
All we have to do to reach the second form is to find a way to rewrite the two logs on the right side as one log. Fortunately the log(1) is zero (since 10 to the zero power is 1) so we can replace that log:
which simplifies to:
We are very close to the second form. The -3 should not be there. Fortunately there is a property of logarithms, , which allows us to move the coefficient of a log into the argument as its exponent. Using this property we can move the -3 out of the way:
which simplifies to:
We now have the second form.
The next step with the second form is based on some simple logic. The equation says that two base 10 logarithms are equal. This means that the exponent on 10 that results in is the same as the exponent on 10 that results in 1/8. Putting a specific exponent on 10 should always result in the same number. So and must be the same since we can use the same exponent on 10 to get them both. So:
Now that the variable is out of the logarithm we can solve for it. Squaring both sides:
Last of all we check the solution. This is not optional! This is required when solving these logarithmic equations or when you square both sides of an equation. We have done both.
Use the original equation to check:
Checking x = 1/64:
Simplifying:
Check!! So x = 1/64 is the solution to the equation.
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