You can put this solution on YOUR website!
When the variable is in an exponent, like in this equation, logarithms are often used to solve it. But not always.
Equations like this can be solved without logarithms if it is possible to rewrite the equation so that both sides are powers of the same base. This equation is such an equation.
The left side is a power of a fraction. The right side is just a fraction. But with some effort we can determine that and so...
Both sides of the equation are now powers. But the bases, 3/4 and 4/3, are not the same. But they are reciprocals of each other. So we can get them to match with a negative exponent:
We now have the equation in the desired form. The equation now says that two powers of 3/4 are equal. The only way this can be true is if the exponents are equal, too. So...
P.S. All equations like this can be solved using logarithms. Only some of them can be solved the way we did above. The method above is faster, easier and more accurate. (Using logarithms often involves use of decimal approximations.) So this method should be preferred. Use logarithms only when you have to.
In case you need to use logarithms, here is a logarithm solution to the problem:
Any base of logarithm may be used. I am using base 10 logs because I want to be able to change it to a decimal approximation easily. Using the property:
Dividing by log(3/4):
This is an exact expression for the solution. Now we get out our calculators: (rounded to 4 places)
Note: Because of the rounding we got an answer that not correct but is very close to the correct answer. This is why the other method, when available, should be the first choice.
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