Question

Find the solution of the exponential equation, correct to four decimal places. 5ˣ = 4ˣ⁺¹

Answer 1

The solution to the exponential equation 5^x = 4^(x+1) involves using natural logarithms and the power rule of logarithms to solve for x.

Step-by-step explanation:

The solution to the exponential equation 5x = 4x + 1 can be found by setting both sides of the equation to the same base or by using logarithms. For this problem, we will use logarithms since the bases are different and there is no simple way to rewrite them to have the same base.

First, take the natural logarithm (ln) of both sides of the equation:

• ln(5x) = ln(4x+1)

Now, apply the power rule of logarithms (logarithm of a power) to move the exponents in front of the logarithms:

• x · ln(5) = (x + 1) · ln(4)

Next, distribute the logarithms:

• x · ln(5) = x · ln(4) + ln(4)

To solve for x, get all terms involving x on one side and constant terms on the other:

• x · ln(5) - x · ln(4) = ln(4)

Factor out x:

• x · (ln(5) - ln(4)) = ln(4)

Divide both sides by (ln(5) - ln(4)) to isolate x:

• x = ln(4)/(ln(5) - ln(4))

Use a calculator to approximate the value of x:

• x ≈ numerical value of ln(4)/(ln(5) - ln(4)), correct to four decimal places

The exact numerical value can be obtained using a scientific calculator or a computational tool.