Question

Consider the following.

e^0.8x = 7
(a) Find the exact solution of the exponential equation in terms of logarithms.
x =



(b) Use a calculator to find an approximation to the solution rounded to six decimal places.
x =

Answer 1

`\textsf{(a)} \quad x=1.25 \ln 7`

`\textsf{(b)} \quad x=2.432388\; \sf (6\; d.p.)`

Explanation:

To find the exact solution of the exponential equation
`e^(0.8x)=7` in terms of logarithms, take the logarithms of both sides of the equation.

We can use logarithms of any base, but the best choice is base e since it is the base of the given exponential expression.

A logarithm with base e is called a natural logarithm and is written as:

`\log_e(x)=\ln(x)`

Part (a)

To find the exact solution of the given exponential equation, take natural logarithms of both sides of the equation:

`\implies \ln e^(0.8x)=\ln 7`

`\textsf{Apply the log power law:} \quad \ln x^n=n \ln x`

`\implies 0.8x \ln e = \ln 7`

As ln(e) = 1:

`\implies 0.8x = \ln 7`

Finally, divide both sides by 0.8:

`\implies x=(1)/(0.8) \ln 7`

`\implies x=1.25 \ln 7`

Therefore, the exact solution is:

• 
  `x = 1.25 \ln 7`

Part (b)

Using a calculator, the solution to six decimal places is:

• 
  `x=2.432388`

Answer 2

(a) x = 1.25 ln(7)

(b) x ≈ 2.432388

Explanation:

(a) To find the exact solution of the equation e^0.8x = 7, we can take the natural logarithm of both sides:

ln(e^0.8x) = ln(7)

Using the property of logarithms that ln(e^a) = a, we can simplify the left side:

0.8x = ln(7)

Finally, solving for x, we get:

x = ln(7) / 0.8

Therefore, the exact solution of the equation is x = ln(7) / 0.8 = 1.25 ln(7)

x = 1.25 ln(7)

(b) To find an approximation of the solution rounded to six decimal places, we can use a calculator to evaluate ln(7) / 0.8:

x = 1.25 * 1.945910149055313 = 2.432387686319142

x ≈ 2.432387686319142

Rounding this to six decimal places, we get:

x ≈ 2.432388

Therefore, an approximation of the solution rounded to six decimal places is x ≈ 2.432388