For this discussion, you will be able to pick one of the learning objectives we have covered in
Module 6 or Module 7 in this case logarithms and develop a step-by-step list on how to solve the
problem generically. Your post will contain all of the following:

State the learning objective that you are covering.
The How-to Guide you created for solving a generic form of the objective in your own words.
Give an example to illustrate the learning objective.

Question

For this discussion, you will be able to pick one of the learning objectives we have covered in
Module 6 or Module 7 in this case logarithms and develop a step-by-step list on how to solve the
problem generically. Your post will contain all of the following:

State the learning objective that you are covering.
The How-to Guide you created for solving a generic form of the objective in your own words.
Give an example to illustrate the learning objective.

UpStudy AI · Accepted Answer

## Learning Objective: Solving Exponential Equations Using Logarithms

### Step-by-Step How-to Guide

1. **Isolate the Exponential Expression**  
   Ensure that the exponential term is by itself on one side of the equation.  
   For example, in an equation of the form  
   $$
   a \cdot b^{cx+d} = e
   $$  
   you first divide by $$a$$ (if $$a \neq 1$$) to isolate the exponential term.

2. **Take the Logarithm of Both Sides**  
   Apply the logarithm (common logarithm $$\log$$ or natural logarithm $$\ln$$) to both sides. This will allow you to "bring down" the exponent using the logarithmic property:
   $$
   \log(b^{cx+d}) = (cx+d)\log(b)
   $$

3. **Apply the Logarithm Power Rule**  
   Use the property that $$\log(b^y) = y \log(b)$$ to rewrite the exponential expression.

4. **Solve for the Variable**  
   After applying the logarithm, you will have a linear equation in the variable $$x$$. Solve for $$x$$ by isolating it using algebraic techniques (such as subtracting or dividing both sides).

5. **Check Your Solution (Optional)**  
   Substitute the solution back into the original equation to verify that both sides are equal.

### Example

Solve the exponential equation:
$$
3^{2x-1} = 7
$$

**Step 1: Isolate the Exponential Expression**  
The exponential term is already isolated.

**Step 2: Take the Logarithm of Both Sides**  
Apply the natural logarithm:
$$
\ln(3^{2x-1}) = \ln(7)
$$

**Step 3: Apply the Logarithm Power Rule**  
Bring the exponent down:
$$
(2x-1)\ln(3) = \ln(7)
$$

**Step 4: Solve for $$x$$**  
Divide both sides of the equation by $$\ln(3)$$:
$$
2x-1 = \frac{\ln(7)}{\ln(3)}
$$  
Now, add $$1$$ to both sides:
$$
2x = \frac{\ln(7)}{\ln(3)} + 1
$$  
Finally, divide by $$2$$:
$$
x = \frac{1}{2}\left(\frac{\ln(7)}{\ln(3)} + 1\right)
$$

This is the solution for $$x$$.

**Step 5: Check Your Solution (Optional)**  
Substitute the value of $$x$$ back into the original equation to confirm that both sides are equal.

The above steps provide a systematic approach to solving exponential equations by using logarithms.
**How to Solve Exponential Equations Using Logarithms** 1. **Isolate the Exponential Term**: Make sure the exponential part is alone on one side. 2. **Take the Logarithm**: Apply the logarithm to both sides to bring down the exponent. 3. **Simplify**: Use logarithm properties to solve for the variable. 4. **Solve for the Variable**: Use algebra to find the value of the variable. 5. **Verify**: Check your answer by plugging it back into the original equation. **Example:** Solve $$3^{2x-1} = 7$$. - Isolate: $$3^{2x-1} = 7$$ - Take Log: $$\ln(3^{2x-1}) = \ln(7)$$ - Simplify: $$(2x-1)\ln(3) = \ln(7)$$ - Solve: $$x = \frac{1}{2}\left(\frac{\ln(7)}{\ln(3)} + 1\right)$$ This method helps solve exponential equations by turning them into simpler algebraic equations.

Upstudy AI Solution

Answer

Solution

Learning Objective: Solving Exponential Equations Using Logarithms

Step-by-Step How-to Guide

Example

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