Question

How to solve the equation step by step ?

`6^(x+3) = 11^x`

Desired answer : `x=8.868`

Answer 1

Given: `6^(x+3) = 11^x`

Use any base logarithm that you like on both sides. I shall use the two logarithms that are on most calculators, the natural logarithm and the base 10 logarithm:

`ln(6^(x+3)) = ln(11^x)`

`log_10(6^(x+3)) = log_10(11^x)`

It is a property of any logarithm that exponentiation within the argument is equal to multiplication of the logarithm by the exponent:

`log_b(a^c) = (c)log_b(a)`

Applying this to the equation:

`(x+3)ln(6) = (x)ln(11)`

`(x+3)log_10(6) = (x)log_10(11)`

Use the distributive property and flip the equation:

`(x)ln(11) = (x)ln(6)+(3)ln(6)`

`(x)log_10(11) = (x)log_10(6)+(3)log_10(6)`

Combine like terms:

`(ln(11) -ln(6))x=(3)ln(6)`

`(log_10(11) -log_10(6))x=(3)log_10(6)`

Divide both sides by the leading coefficient:

`x=((3)ln(6))/(ln(11) -ln(6))`

`x=((3)log_10(6))/(log_10(11) -log_10(6))`

Using a calculator:

`x~~ 8.86810`