Write answer using base-10 logarithms

Question

Solve for X.

 

2^x-6 = 7^2x

 

Write the exact answer using base-10 logarithms.

 

Help Please.

Answer 1

2^x-6 = 7^2x

log 2^x-6 = log 7^2x

(x - 6)(log 2) = 2x(log 7)

(x)(log 2) - 6(log 2) = 2x(log 7)

-6(log 2) = 2x(log 7) - xlog 2

-6(log2) = x[2log 7 - log 2]

(-6 log 2) / (2log 7 - log 2] = x

 

This is the exact value.

 

Answer 2

This is a tricky problem! Your first step is to remember that a^y = b is the same thing as log_ab = y. For your problem, let a = 2. Then b = 7^2x and y = x - 6. So you have log₂(7^2x) = x - 6. That may look equally daunting, but here you can make use of the rule that tells you log_a(b^x) = xlog_ab. In your case, this means we have 2x(log₂7) = x - 6.

 

From this point it becomes a regular algebra problem. How would you solve this problem if it were 2xk = x - 6, where k is some constant? Let me know if you need further help.