Solve for x. Write the exact answer using base-10.

Question

2^6x=15^-x+2

Answer 1

Since the answer is supposed to be given in terms of log base 10 (common log), start by taking common log of both sides (this is the strategy when the variable to solve for is in an exponent).

log(2^6x) = log(15^(-x+2)

Apply the power property of logarithms to bring the exponents in front of the logs:

6xlog2 = (-x+2)log15

Distribute the log15 on right side of equation:

6xlog2 = -xlog15 + 2log15

Bring the terms containing x to the same side of the equal sign:

6xlog2 + xlog15 = 2log15

Factor the x from both terms on the left:

x(6log2+log15) = 2log15

Divide both sides by the coefficient of x:

x = 2log15/(6log2+log15)

You could use properties of logarithms to rewrite this as:

x= log(225)/log(960)

Which you could also write as"

x = log₉₆₀(225)

For confirmation and a decimal approximation for the solution:

desmos.com/calculator/zb38yohq2j