
Question text
Solve lne2x−1
 =5

Question 10Select one:

a.
ln (x-1)


b.
2x-1(ln 1)


c.
52


d.
3



To solve the equation:

\[
\ln(2x - 1) = 5
\]

### Step 1: Remove the natural logarithm
We know that the natural logarithm function is the inverse of the exponential function. To eliminate the logarithm, exponentiate both sides of the equation with base \( e \):

\[
2x - 1 = e^5
\]

### Step 2: Solve for \( x \)
Now solve for \( x \):

\[
2x = e^5 + 1
\]

\[
x = \frac{e^5 + 1}{2}
\]

So the solution is:

\[
x = \frac{e^5 + 1}{2}
\]

From the answer choices, none exactly match this value, so it appears that this problem is not multiple-choice solvable directly. However, it's possible you were expected to choose a closely related simplification or consider the steps rather than an exact numerical answer.

Would you like further clarification or more details on why the choices don't fit the solution?

### Expansion Questions:
1. How can the properties of logarithms be applied to simplify more complex equations?
2. What is the inverse of a natural logarithm function, and how is it used in algebra?
3. What does \( e^5 \) represent in terms of approximate value, and when is it useful to approximate in calculations?
4. Why is it necessary to add \( 1 \) in the process of solving \( 2x - 1 = e^5 \)?
5. How would you solve \( \ln(2x + 3) = 4 \)?

### Tip:
Always check whether the logarithmic equation gives a valid result (e.g., the argument of a logarithm must be positive).

This detailed solution explains how to solve the logarithmic equation ln(2x - 1) = 5 by using the inverse properties of logarithms and exponentials. Key algebraic concepts and formulas are applied, such as exponentiation and solving for x. Suitable for high school students in Grades 10-12.

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