Math Problem Statement

Solve the equation 4^x = 21.

Solution

The equation presented in the image is:

4x=214^x = 21

To solve for xx, we can use logarithms.

Solution

  1. Take the natural logarithm (or common logarithm) of both sides: ln(4x)=ln(21)\ln(4^x) = \ln(21)

  2. Use the power rule for logarithms, which states ln(ab)=bln(a)\ln(a^b) = b \ln(a): xln(4)=ln(21)x \cdot \ln(4) = \ln(21)

  3. Isolate xx by dividing both sides by ln(4)\ln(4): x=ln(21)ln(4)x = \frac{\ln(21)}{\ln(4)}

  4. Calculate the value using approximate values for the natural logarithms: x3.04451.38632.195x \approx \frac{3.0445}{1.3863} \approx 2.195

Answer:

x2.195x \approx 2.195

Would you like a deeper explanation or have any questions?

Further Questions to Explore

  1. What is the effect of changing the base (4 in this case) on the solution of the equation?
  2. How would the solution change if the equation were 4x=404^x = 40?
  3. Can we solve this equation using a different logarithm base, such as base 10?
  4. How do logarithmic functions relate to exponential functions like 4x4^x?
  5. What are some real-life applications of solving exponential equations like this one?

Tip:

When solving exponential equations, using logarithms is often the most straightforward method to isolate the variable in the exponent.

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Math Problem Analysis

Mathematical Concepts

Algebra
Exponential Equations
Logarithms

Formulas

Power rule of logarithms: ln(a^b) = b * ln(a)
Change of base formula for logarithms

Theorems

Properties of logarithms

Suitable Grade Level

Grades 10-12