Math Problem Statement

Explain how to solve 3^x - 4 = 6 using the change of base formula log_b(y) = log_k(y) / log_k(b). Include the solution for x in your answer. Round your answer to the nearest thousandth.

Solution

To solve the equation 3x4=63^x - 4 = 6 using the change of base formula, we will proceed step-by-step:

Step 1: Isolate the Exponential Expression

Start by isolating the term involving the exponential expression:

3x4=63^x - 4 = 6

Add 4 to both sides:

3x=103^x = 10

Step 2: Apply Logarithm to Both Sides

To solve for xx, take the logarithm of both sides. We can use any logarithm, but for consistency, let's use the natural logarithm (base ee):

ln(3x)=ln(10)\ln(3^x) = \ln(10)

Step 3: Use the Logarithm Power Rule

Apply the power rule of logarithms, which states ln(ab)=bln(a)\ln(a^b) = b \ln(a):

xln(3)=ln(10)x \ln(3) = \ln(10)

Step 4: Solve for xx

Now, solve for xx by dividing both sides by ln(3)\ln(3):

x=ln(10)ln(3)x = \frac{\ln(10)}{\ln(3)}

Step 5: Use the Change of Base Formula

The change of base formula for logarithms states that:

logb(y)=logk(y)logk(b)\log_b(y) = \frac{\log_k(y)}{\log_k(b)}

In this case, using natural logarithms (ln\ln) or common logarithms (log\log) provides the same solution. Therefore:

x=log(10)log(3)x = \frac{\log(10)}{\log(3)}

Step 6: Calculate the Value of xx

Now, compute the value using a calculator:

xlog(10)log(3)10.47712.096x \approx \frac{\log(10)}{\log(3)} \approx \frac{1}{0.4771} \approx 2.096

Step 7: Round to the Nearest Thousandth

Rounded to the nearest thousandth, the value of xx is:

x2.096x \approx 2.096

Final Answer

The solution for xx is approximately 2.0962.096.

Would you like more details or have any questions?

Related Questions:

  1. How do you solve exponential equations that involve different bases?
  2. Can you solve 2x+5=132^x + 5 = 13 using a logarithm?
  3. What is the power rule of logarithms, and how is it applied?
  4. How does the change of base formula work for logarithms other than base 10 or ee?
  5. What are the differences between natural logarithms (ln\ln) and common logarithms (log\log)?

Tip:

When solving equations involving exponents, always consider using logarithms to simplify and isolate the variable.

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Math Problem Analysis

Mathematical Concepts

Algebra
Exponential Equations
Logarithms

Formulas

Change of Base Formula: log_b(y) = log_k(y) / log_k(b)
Logarithm Power Rule: ln(a^b) = b * ln(a)

Theorems

Change of Base Theorem
Logarithm Power Rule

Suitable Grade Level

Grades 10-12