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If the logarithm was a base 5 logarithm the expression on the left would simplify easily. This is true because of what logarithms are. Logarithms are exponents, Base 5 logarithms are exponents for a 5.
is the exponent for 5 that results in q. So if we had an expression:
it would simplify to 4x because
is the exponent for 5 that results in 4x and we find
as the exponent for a 5!
So we are going to start by using the base conversion formula,
, to convert the base 25 logarithm into an expression of base 5 logarithms:
represents the exponent for 5 that results in 25 and since we know that the exponent for 5 that results in 25 is 3, this simplifies to:
We are now close the the easily simplified 5 raised to the base 5 logarithm power we have been trying to get. The 2 needs to go. Here's how. Dividing by two is the same as multiplying by 1/2:
Now we can use a property of logarithms,
, to move the 1/2 into the argument as an exponent:
The left side is now in the easily simplified form. Since
represents the exponent for 5 that results in
is the exponent on a 5, the left side must be
. This gives us:
We can now solve for x. Square both sides:
4x = 64
Divide by 4:
x = 16.