Solve each exponential equation by converting to a common base:

Question

Show step by step work:

16×2^7x-36=8^13x

Answer 1

The strategy here is to get all the numbers to have a common base. In this case:

16 can be written as 2⁴

8 can be written as 2³

So we can write all the numbers with a base of 2.

1) When two numbers, having the same base and having exponents, are multiplied, we can add their exponents:

b^m • bⁿ = b^m+n Example: 2⁵ • 2⁴ = 2⁹

So 16 • 2^7x-36 can be written as 2⁴ • 2^{7x - 36} or 2^{7x - 36 + 4} or 2^{7x - 32}

2) When a number with an exponent is raised to an exponent, then we can multiply the exponent:

(b^m)ⁿ = b^m•n Example: (2⁵)⁴ = 2²⁰

So 8^13x can be written as (2³)^13x or 2^3•13x or 2^39x

Putting those together we get:

16 • 2^7x-36 = 8^13x

2⁴ • 2^{7x - 36} = (2³)^13x

2^{7x - 32} = 2^39x

Since the bases are the same, then the exponents must be equal:

7x - 32 = 39x

-32 = 32x

x = -1