Math Problem Statement

^{\placeholder{}} $$$$ 3^{x+1}=\exponentialE^7

Solution

You want to solve the equation 3x+1=e73^{x+1} = e^7.

Step-by-Step Solution:

  1. Start with the equation: 3x+1=e73^{x+1} = e^7

  2. Apply the natural logarithm (ln) to both sides to simplify the exponents: ln(3x+1)=ln(e7)\ln(3^{x+1}) = \ln(e^7)

  3. Use the logarithmic identity ln(ab)=bln(a)\ln(a^b) = b \ln(a) to simplify: (x+1)ln(3)=7(x+1) \ln(3) = 7

  4. Solve for x+1x+1: x+1=7ln(3)x+1 = \frac{7}{\ln(3)}

  5. Finally, solve for xx by subtracting 1: x=7ln(3)1x = \frac{7}{\ln(3)} - 1

This is the solution for xx.

Would you like a detailed explanation or clarification on any step?


Here are 5 related questions to consider:

  1. How do logarithmic identities like ln(ab)=bln(a)\ln(a^b) = b \ln(a) work in different contexts?
  2. What is the significance of the natural logarithm in solving exponential equations?
  3. How does base ee (Euler's number) differ from other bases like 3 or 10 in logarithmic functions?
  4. How can you graph functions like 3x+13^{x+1} and e7e^7 to visually find the solution for xx?
  5. What methods exist to approximate solutions when logarithmic expressions are involved?

Tip: When solving exponential equations, applying logarithms can simplify exponents into linear equations, making them much easier to handle.

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Math Problem Analysis

Mathematical Concepts

Algebra
Exponential Equations
Logarithms

Formulas

Natural logarithm: ln(a^b) = b * ln(a)
Exponential equation: 3^{x+1} = e^7
Solution for x in exponential equations: x = (7 / ln(3)) - 1

Theorems

Logarithmic identity: ln(a^b) = b * ln(a)
Properties of logarithms: ln(e^x) = x

Suitable Grade Level

Grades 10-12