Math Problem Statement

Solve 4^(5x-2) = 3^(-2x+1), giving the answer in the form x = ln(a)/ln(b)

Solution

The equation given is:

45x2=32x+14^{5x - 2} = 3^{-2x + 1}

You are required to solve for xx and express the answer in the form x=lnalnbx = \frac{\ln a}{\ln b}.

Step 1: Apply the natural logarithm to both sides

We can take the natural logarithm (ln\ln) of both sides to deal with the exponents.

ln(45x2)=ln(32x+1)\ln(4^{5x - 2}) = \ln(3^{-2x + 1})

Step 2: Use the logarithmic power rule

The logarithmic power rule states that ln(ab)=bln(a)\ln(a^b) = b \cdot \ln(a). Apply this rule to both sides:

(5x2)ln(4)=(2x+1)ln(3)(5x - 2) \cdot \ln(4) = (-2x + 1) \cdot \ln(3)

Step 3: Expand both sides

Distribute the logarithms across the terms:

(5x)ln(4)2ln(4)=(2x)ln(3)+ln(3)(5x) \cdot \ln(4) - 2 \cdot \ln(4) = (-2x) \cdot \ln(3) + \ln(3)

Step 4: Group like terms

Now, group all the terms involving xx on one side and constants on the other side.

(5x)ln(4)+(2x)ln(3)=ln(3)+2ln(4)(5x) \cdot \ln(4) + (2x) \cdot \ln(3) = \ln(3) + 2 \cdot \ln(4)

Step 5: Factor out xx

Factor out xx on the left-hand side:

x(5ln(4)+2ln(3))=ln(3)+2ln(4)x \left( 5 \cdot \ln(4) + 2 \cdot \ln(3) \right) = \ln(3) + 2 \cdot \ln(4)

Step 6: Solve for xx

Now, solve for xx by dividing both sides by (5ln(4)+2ln(3))(5 \cdot \ln(4) + 2 \cdot \ln(3)):

x=ln(3)+2ln(4)5ln(4)+2ln(3)x = \frac{\ln(3) + 2 \cdot \ln(4)}{5 \cdot \ln(4) + 2 \cdot \ln(3)}

Final Answer:

This is the value of xx in the required form:

x=ln(3)+2ln(4)5ln(4)+2ln(3)x = \frac{\ln(3) + 2 \cdot \ln(4)}{5 \cdot \ln(4) + 2 \cdot \ln(3)}

Let me know if you need further clarification or details!


Would you like to explore more details on this solution? Here are 5 related questions:

  1. How do logarithmic rules apply when solving exponential equations?
  2. Why do we use natural logarithms specifically in this case?
  3. Can the solution be simplified further or approximated?
  4. How would the solution differ if the bases were the same on both sides of the equation?
  5. What real-world problems involve solving equations of this form?

Tip: When solving logarithmic equations, always ensure the base is valid (i.e., greater than zero and not equal to one) for the natural logarithm function.

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Math Problem Analysis

Mathematical Concepts

Exponential Equations
Logarithms
Algebra

Formulas

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

Theorems

Logarithmic Power Rule

Suitable Grade Level

Grades 10-12