Problem 1: Logarithmic Equation

We start with:
$\log(3) + \log(x - 8) = \log(x)$

(a) Combine the logarithms on the LHS
Using the logarithmic property $\log(a) + \log(b) = \log(a \cdot b)$, we combine:
$\log(3(x - 8)) = \log(x)$

(b) Use the one-to-one property
Since the logarithmic function is one-to-one, we equate the arguments:
$3(x - 8) = x$

(c) Solve for $x$:
Expanding and simplifying:
$3x - 24 = x \quad \Rightarrow \quad 3x - x = 24 \quad \Rightarrow \quad 2x = 24 \quad \Rightarrow \quad x = 12$

Check for extraneous solutions:
Substitute $x = 12$ into the original equation:
$\log(3) + \log(12 - 8) = \log(12)$ $\log(3) + \log(4) = \log(12)$ $\log(12) = \log(12) \quad \text{(True!)}$

Thus, the solution is:
$x = 12$

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Problem 2: Exponential Equation

We start with:
$3^{2x - 1} = 1$

Since $1 = 3^0$, equate the exponents:
$2x - 1 = 0$ $2x = 1 \quad \Rightarrow \quad x = \frac{1}{2}$

Thus, the solution is:
$x = \frac{1}{2}$

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Problem 3: Exponential Equation with Base $e$

We start with:
$e^{1 - 3x} = e^{4x - 4}$

Use the one-to-one property of exponential functions to equate exponents:
$1 - 3x = 4x - 4$

Simplify:
$1 + 4 = 4x + 3x \quad \Rightarrow \quad 5 = 7x \quad \Rightarrow \quad x = \frac{5}{7}$

Thus, the solution is:
$x = \frac{5}{7}$

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Problem 4: Exponential Equation with Logarithmic Solution

We start with:
$5^{4x - 1} = 8$

(a) Solve in terms of logarithms:
Take the natural logarithm ($\ln$) on both sides:
$\ln(5^{4x - 1}) = \ln(8)$

Using the logarithmic power rule ($\ln(a^b) = b \ln(a)$):
$(4x - 1) \ln(5) = \ln(8)$

Divide through by $\ln(5)$:
$4x - 1 = \frac{\ln(8)}{\ln(5)}$

Solve for $x$:
$4x = \frac{\ln(8)}{\ln(5)} + 1 \quad \Rightarrow \quad x = \frac{\frac{\ln(8)}{\ln(5)} + 1}{4}$

(b) Approximation:
Using a calculator:
$\ln(8) \approx 2.0794, \quad \ln(5) \approx 1.6094$ $x \approx \frac{\frac{2.0794}{1.6094} + 1}{4} \approx \frac{1.292 + 1}{4} \approx \frac{2.292}{4} \approx 0.573$

Thus, the solution is:
$x \approx 0.573$

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Questions for Further Practice:

1. How do you verify solutions to logarithmic equations?
2. What is the relationship between exponential and logarithmic equations?
3. How can extraneous solutions arise in logarithmic problems?
4. Can you solve $\ln(x) + \ln(x - 1) = 1$?
5. What are the key steps in solving $2^{x+1} = 8$?

Tip: Always check for domain restrictions in logarithmic equations to avoid extraneous solutions!