10^7x-10 = 4^2x-7

Question

I couldn't figure it out. Please help!

Answer 1

In order to solve exponential problems, we must use logarithms.

10^7x-10 = 4^2x-7

Take the logarithm of both sides (let's use log base 10)

log₁₀10^7x-10 = log₁₀4^2x-7

One of the rules of logarithms is that we can move the exponent out in front as a multiplier

(log_bmⁿ = nlog_bm)

So using this rule we get:

(7x-10)log₁₀10 = (2x-7)log₁₀4

Another rule of logarithms is that log_bb = 1 so using this rule, we get:

(7x-10)(1) = (2x-7)log₁₀4

7x - 10 = (2x)log₁₀4 - 7log₁₀4

7x = (2x)log₁₀4 - 7log₁₀4 + 10

7x - (2x)log₁₀4 = -7log₁₀4 + 10

x(7 - 2log₁₀4) = -7log₁₀4 + 10

x = (-7log₁₀4 + 10)/(7 - 2log₁₀4)

Using a calculator, x = 0.9982228832