SOLUTION: 5 raised to log25(4X)=8

Question 387862: 5 raised to log25(4X)=8
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

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:

Since 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 and since 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.
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