Math Problem Statement

Solve for x: 5^x = 26. You may enter the exact value or round to 4 decimal places.

Solution

To solve the equation 5x=265^x = 26, we can use logarithms:

  1. Take the natural logarithm (ln) of both sides: ln(5x)=ln(26)\ln(5^x) = \ln(26)

  2. Apply the logarithmic rule ln(ab)=bln(a)\ln(a^b) = b\ln(a): xln(5)=ln(26)x \ln(5) = \ln(26)

  3. Solve for xx: x=ln(26)ln(5)x = \frac{\ln(26)}{\ln(5)}

Now, calculate the value using a calculator:

x=ln(26)ln(5)3.25811.60942.0248x = \frac{\ln(26)}{\ln(5)} \approx \frac{3.2581}{1.6094} \approx 2.0248

Thus, the solution is: x2.0248(rounded to 4 decimal places).x \approx 2.0248 \, (\text{rounded to 4 decimal places}).

Would you like a more detailed explanation of logarithms or related concepts?


Here are five follow-up questions to explore this topic further:

  1. How does the base of the logarithm affect the solution?
  2. Can we solve the same equation using a different base for the logarithm (e.g., base 10)?
  3. What is the significance of logarithms in exponential growth problems?
  4. How would the solution change if the equation were 5x+1=265^{x+1} = 26?
  5. What practical applications involve solving exponential equations like this one?

Tip: When solving equations involving exponents, always remember logarithmic properties to simplify calculations efficiently.

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Math Problem Analysis

Mathematical Concepts

Exponents
Logarithms

Formulas

log(a^b) = b * log(a)
x = ln(y)/ln(a)

Theorems

Logarithmic properties

Suitable Grade Level

Grades 10-12