QuestionSolve for $x$ using the natural logarithm: $10^{4 x-3}=9$

Studdy Solution

STEP 1

1. The equation $10^{4x - 3} = 9$ is exponential.
2. We will use logarithms to solve for $x$ .

STEP 2

1. Take the natural logarithm of both sides.
2. Use logarithmic properties to simplify the equation.
3. Solve for $x$ .

STEP 3

Take the natural logarithm of both sides of the equation:
$\ln(10^{4x - 3}) = \ln(9)$

STEP 4

Use the property of logarithms that allows you to bring the exponent down in front of the logarithm:
$(4x - 3) \cdot \ln(10) = \ln(9)$

STEP 5

Solve for $x$ by first isolating the term with $x$ :
$4x - 3 = \frac{\ln(9)}{\ln(10)}$
Add 3 to both sides:
$4x = \frac{\ln(9)}{\ln(10)} + 3$
Divide both sides by 4 to solve for $x$ :
$x = \frac{1}{4} \left( \frac{\ln(9)}{\ln(10)} + 3 \right)$
The value of $x$ is:
$x = \frac{1}{4} \left( \frac{\ln(9)}{\ln(10)} + 3 \right)$

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms