Answer :
To evaluate [tex]\(\log _{ a } 147\)[/tex] using the given values [tex]\(\log _{ a } 3 \approx 0.477\)[/tex] and [tex]\(\log _{ a } 7 \approx 0.845\)[/tex], we can utilize the properties of logarithms.
Here's a step-by-step explanation:
1. Factorize the Number 147:
First, recognize that 147 can be expressed as a product of its prime factors:
[tex]\[
147 = 3 \times 49
\][/tex]
Since 49 is [tex]\(7 \times 7\)[/tex], we can rewrite it as:
[tex]\[
147 = 3 \times 7^2
\][/tex]
2. Use Logarithm Properties:
We use the property of logarithms that states:
[tex]\[
\log_b (xy) = \log_b x + \log_b y
\][/tex]
This allows us to write:
[tex]\[
\log _a 147 = \log _a (3 \times 7^2)
\][/tex]
[tex]\[
\log _a 147 = \log _a 3 + \log _a 7^2
\][/tex]
3. Apply the Power Rule:
The power rule for logarithms tells us:
[tex]\[
\log_b (y^c) = c \cdot \log_b y
\][/tex]
Applying this rule:
[tex]\[
\log _a 7^2 = 2 \cdot \log _a 7
\][/tex]
4. Put It All Together:
Now substitute the values you know:
[tex]\[
\log_a 147 = \log_a 3 + 2 \cdot \log_a 7
\][/tex]
Plug in the given values:
[tex]\[
\log_a 3 \approx 0.477
\][/tex]
[tex]\[
\log_a 7 \approx 0.845
\][/tex]
5. Calculate:
[tex]\[
\log_a 147 = 0.477 + 2 \times 0.845
\][/tex]
[tex]\[
\log_a 147 = 0.477 + 1.69
\][/tex]
[tex]\[
\log_a 147 \approx 2.167
\][/tex]
So, the approximate value of [tex]\(\log _a 147\)[/tex] is [tex]\(2.167\)[/tex].
Here's a step-by-step explanation:
1. Factorize the Number 147:
First, recognize that 147 can be expressed as a product of its prime factors:
[tex]\[
147 = 3 \times 49
\][/tex]
Since 49 is [tex]\(7 \times 7\)[/tex], we can rewrite it as:
[tex]\[
147 = 3 \times 7^2
\][/tex]
2. Use Logarithm Properties:
We use the property of logarithms that states:
[tex]\[
\log_b (xy) = \log_b x + \log_b y
\][/tex]
This allows us to write:
[tex]\[
\log _a 147 = \log _a (3 \times 7^2)
\][/tex]
[tex]\[
\log _a 147 = \log _a 3 + \log _a 7^2
\][/tex]
3. Apply the Power Rule:
The power rule for logarithms tells us:
[tex]\[
\log_b (y^c) = c \cdot \log_b y
\][/tex]
Applying this rule:
[tex]\[
\log _a 7^2 = 2 \cdot \log _a 7
\][/tex]
4. Put It All Together:
Now substitute the values you know:
[tex]\[
\log_a 147 = \log_a 3 + 2 \cdot \log_a 7
\][/tex]
Plug in the given values:
[tex]\[
\log_a 3 \approx 0.477
\][/tex]
[tex]\[
\log_a 7 \approx 0.845
\][/tex]
5. Calculate:
[tex]\[
\log_a 147 = 0.477 + 2 \times 0.845
\][/tex]
[tex]\[
\log_a 147 = 0.477 + 1.69
\][/tex]
[tex]\[
\log_a 147 \approx 2.167
\][/tex]
So, the approximate value of [tex]\(\log _a 147\)[/tex] is [tex]\(2.167\)[/tex].
