Use the laws of logarithms to rewrite the expressions into a single logarithm.

(a) log5 (*x* 2 *−* 1) *−* log5 (*x −* 1)

(b) ln(*a* + *b*) + ln(*a − b*) *−* 2 ln(*c*)

(c) 2(log5 (*x*) + 2 log5 (*y*) *−* 3 log5 (*z*))

(d)

1

3 log[(*x* + 2)3 ] + 1 2  log

To rewrite each expression into a single logarithm, we’ll use the laws of logarithms, specifically:

1. **Product Rule**: \( \log_b(M) + \log_b(N) = \log_b(M \cdot N) \)
2. **Quotient Rule**: \( \log_b(M) - \log_b(N) = \log_b\left(\frac{M}{N}\right) \)
3. **Power Rule**: \( n \cdot \log_b(M) = \log_b(M^n) \)

Let's solve each problem.

---

### (a) \(\log_5 (x^2 - 1) - \log_5 (x - 1)\)

Using the **Quotient Rule**:

\[
\log_5 (x^2 - 1) - \log_5 (x - 1) = \log_5 \left( \frac{x^2 - 1}{x - 1} \right)
\]

Since \( x^2 - 1 \) can be factored as \((x + 1)(x - 1)\):

\[
\log_5 \left( \frac{(x + 1)(x - 1)}{x - 1} \right) = \log_5 (x + 1)
\]

So, the simplified form is:

\[
\log_5 (x + 1)
\]

---

### (b) \(\ln(a + b) + \ln(a - b) - 2 \ln(c)\)

First, apply the **Product Rule** to combine the first two terms:

\[
\ln(a + b) + \ln(a - b) = \ln((a + b)(a - b))
\]

Now, use the **Power Rule** for the last term:

\[
2 \ln(c) = \ln(c^2)
\]

Now we have:

\[
\ln((a + b)(a - b)) - \ln(c^2)
\]

Finally, apply the **Quotient Rule**:

\[
\ln \left( \frac{(a + b)(a - b)}{c^2} \right)
\]

So, the simplified form is:

\[
\ln \left( \frac{(a + b)(a - b)}{c^2} \right)
\]

---

### (c) \(2(\log_5 (x) + 2 \log_5 (y) - 3 \log_5 (z))\)

Distribute the \(2\) to each term inside the parentheses using the **Power Rule**:

\[
2 \log_5 (x) + 4 \log_5 (y) - 6 \log_5 (z)
\]

Now apply the **Power Rule** to each term:

\[
\log_5 (x^2) + \log_5 (y^4) - \log_5 (z^6)
\]

Next, use the **Product Rule** for the first two terms and then the **Quotient Rule**:

\[
\log_5 \left( x^2 \cdot y^4 \right) - \log_5 (z^6) = \log_5 \left( \frac{x^2 y^4}{z^6} \right)
\]

So, the simplified form is:

\[
\log_5 \left( \frac{x^2 y^4}{z^6} \right)
\]

---

### (d) \(\frac{1}{3} \log((x + 2)^3) + \frac{1}{2} \log(y)\)

Use the **Power Rule** for each term:

\[
\frac{1}{3} \log((x + 2)^3) = \log((x + 2))
\]
\[
\frac{1}{2} \log(y

Use the laws of logarithms to rewrite the expressions into a single logarithm.

(a) log5 (x^2 − 1) − log5 (x − 1)

(b) ln(a + b) + ln(a − b) − 2 ln(c)

(c) 2(log5 (x) + 2 log5 (y) − 3 log5 (z))

(d) 1/3 log((x + 2)^3) + 1/2 log(y)

Learn how to simplify logarithmic expressions using the laws of logarithms. This guide covers the product, quotient, and power rules, with detailed solutions for combining multiple logarithmic terms into a single logarithmic expression. Suitable for high school students in Grades 10-12.

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