We are tasked with solving the equation:

$4^{2x - 1} = 5^{x + 2}$

Step 1: Rewrite the equation

First, we rewrite $4^{2x - 1}$ as $(2^2)^{2x - 1}$, which simplifies to:

$(2^2)^{2x - 1} = 2^{4x - 2}$

So the equation becomes:

$2^{4x - 2} = 5^{x + 2}$

Step 2: Take the logarithm of both sides

To solve for $x$, we take the logarithm (base 10 or natural log) of both sides. Using natural logarithm (ln):

$\ln(2^{4x - 2}) = \ln(5^{x + 2})$

Step 3: Use the logarithmic property

Apply the logarithmic property $\ln(a^b) = b \ln(a)$:

$(4x - 2) \ln(2) = (x + 2) \ln(5)$

Step 4: Expand and collect terms

Now, expand both sides:

$(4x - 2) \ln(2) = x \ln(5) + 2 \ln(5)$

Distribute:

$4x \ln(2) - 2 \ln(2) = x \ln(5) + 2 \ln(5)$

Step 5: Collect $x$-terms on one side

Move all terms involving $x$ to one side and constants to the other:

$4x \ln(2) - x \ln(5) = 2 \ln(5) + 2 \ln(2)$

Factor out $x$ on the left-hand side:

$x(4 \ln(2) - \ln(5)) = 2 \ln(5) + 2 \ln(2)$

Step 6: Solve for $x$

Now, solve for $x$ by dividing both sides:

$x = \frac{2 \ln(5) + 2 \ln(2)}{4 \ln(2) - \ln(5)}$

This expression gives the exact solution for $x$. You can approximate the value using logarithms:

$\ln(2) \approx 0.6931, \quad \ln(5) \approx 1.6094$

Substitute these values:

$x \approx \frac{2(1.6094) + 2(0.6931)}{4(0.6931) - 1.6094}$

Simplifying:

$x \approx \frac{3.2188 + 1.3862}{2.7724 - 1.6094} = \frac{4.605}{1.163} \approx 3.96$

Thus, the approximate solution is:

$x \approx 3.96$

Would you like further details or clarification?

Related Questions:

1. How do you solve logarithmic equations more generally?
2. What are the properties of logarithms used in solving exponential equations?
3. How would the approach change if the bases were the same on both sides of the equation?
4. Can logarithms be used to solve any exponential equation, or are there exceptions?
5. How can we simplify logarithmic expressions more efficiently?

Tip:

Always remember that for exponential equations, logarithms are your key tool to transform and solve them when you cannot make the bases identical.