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Sagot :
Let's solve each part step-by-step.
### (i) Solve the equation [tex]\( 16 \log_8 4 = \log_4 r \)[/tex]
1. Rewrite [tex]\( \log_8 4 \)[/tex]:
Use the change of base formula for logarithms, which states [tex]\(\log_b a = \frac{\log_k a}{\log_k b}\)[/tex] for any base [tex]\(k\)[/tex].
[tex]\[ \log_8 4 = \frac{\log 4}{\log 8} \][/tex]
2. Simplify the logarithms:
Recognize that [tex]\(8 = 2^3\)[/tex] and [tex]\(4 = 2^2\)[/tex].
[tex]\[ \log 8 = \log(2^3) = 3 \log 2 \][/tex]
[tex]\[ \log 4 = \log(2^2) = 2 \log 2 \][/tex]
Substitute these values back in:
[tex]\[ \log_8 4 = \frac{2 \log 2}{3 \log 2} = \frac{2}{3} \][/tex]
3. Multiply by 16:
[tex]\[ 16 \log_8 4 = 16 \cdot \frac{2}{3} = \frac{32}{3} \][/tex]
4. Rewrite the equation:
Now we have:
[tex]\[ \frac{32}{3} = \log_4 r \][/tex]
5. Convert to exponential form:
[tex]\[ 4^{\frac{32}{3}} = r \][/tex]
6. Simplify:
Recognize that [tex]\(4 = 2^2\)[/tex], so:
[tex]\[ 4^{\frac{32}{3}} = (2^2)^{\frac{32}{3}} = 2^{\frac{64}{3}} \][/tex]
Thus,
[tex]\[ r = 2^{\frac{64}{3}} \][/tex]
### (ii) Solve the equation [tex]\( \log_5 9 + \log_5 12 + \log_5 15 + \log_5 18 = 1 + \log_5 x + \log_5 x^2 \)[/tex]
1. Combine the sums of logarithms using the product rule:
[tex]\[ \log_5 9 + \log_5 12 + \log_5 15 + \log_5 18 = \log_5 (9 \times 12 \times 15 \times 18) \][/tex]
2. Calculate the product:
Compute the product inside the logarithm:
[tex]\[ 9 \times 12 = 108 \][/tex]
[tex]\[ 108 \times 15 = 1620 \][/tex]
[tex]\[ 1620 \times 18 = 29160 \][/tex]
Thus,
[tex]\[ \log_5 (9 \times 12 \times 15 \times 18) = \log_5 29160 \][/tex]
3. Simplify the right-hand side:
Combine the logarithms on the right:
[tex]\[ 1 + \log_5 x + \log_5 x^2 = 1 + \log_5 (x \cdot x^2) = 1 + \log_5 x^3 \][/tex]
4. Convert 1 to a logarithm:
Since [tex]\(\log_5 5 = 1\)[/tex],
[tex]\[ 1 + \log_5 x^3 = \log_5 5 + \log_5 x^3 = \log_5 (5 \cdot x^3) \][/tex]
5. Set the logarithms equal to each other:
[tex]\[ \log_5 29160 = \log_5 (5x^3) \][/tex]
6. Remove the logarithms:
[tex]\[ 29160 = 5x^3 \][/tex]
7. Solve for [tex]\(x^3\)[/tex]:
[tex]\[ x^3 = \frac{29160}{5} = 5832 \][/tex]
8. Solve for [tex]\(x\)[/tex]:
Take the cube root of both sides:
[tex]\[ x = \sqrt[3]{5832} \][/tex]
Note that [tex]\(5832 = 18^3\)[/tex], so:
[tex]\[ x = 18 \][/tex]
Thus, the solution for the second equation is:
[tex]\[ x = 18 \][/tex]
### Summary
1. [tex]\( r = 2^{\frac{64}{3}} \)[/tex]
2. [tex]\( x = 18 \)[/tex]
### (i) Solve the equation [tex]\( 16 \log_8 4 = \log_4 r \)[/tex]
1. Rewrite [tex]\( \log_8 4 \)[/tex]:
Use the change of base formula for logarithms, which states [tex]\(\log_b a = \frac{\log_k a}{\log_k b}\)[/tex] for any base [tex]\(k\)[/tex].
[tex]\[ \log_8 4 = \frac{\log 4}{\log 8} \][/tex]
2. Simplify the logarithms:
Recognize that [tex]\(8 = 2^3\)[/tex] and [tex]\(4 = 2^2\)[/tex].
[tex]\[ \log 8 = \log(2^3) = 3 \log 2 \][/tex]
[tex]\[ \log 4 = \log(2^2) = 2 \log 2 \][/tex]
Substitute these values back in:
[tex]\[ \log_8 4 = \frac{2 \log 2}{3 \log 2} = \frac{2}{3} \][/tex]
3. Multiply by 16:
[tex]\[ 16 \log_8 4 = 16 \cdot \frac{2}{3} = \frac{32}{3} \][/tex]
4. Rewrite the equation:
Now we have:
[tex]\[ \frac{32}{3} = \log_4 r \][/tex]
5. Convert to exponential form:
[tex]\[ 4^{\frac{32}{3}} = r \][/tex]
6. Simplify:
Recognize that [tex]\(4 = 2^2\)[/tex], so:
[tex]\[ 4^{\frac{32}{3}} = (2^2)^{\frac{32}{3}} = 2^{\frac{64}{3}} \][/tex]
Thus,
[tex]\[ r = 2^{\frac{64}{3}} \][/tex]
### (ii) Solve the equation [tex]\( \log_5 9 + \log_5 12 + \log_5 15 + \log_5 18 = 1 + \log_5 x + \log_5 x^2 \)[/tex]
1. Combine the sums of logarithms using the product rule:
[tex]\[ \log_5 9 + \log_5 12 + \log_5 15 + \log_5 18 = \log_5 (9 \times 12 \times 15 \times 18) \][/tex]
2. Calculate the product:
Compute the product inside the logarithm:
[tex]\[ 9 \times 12 = 108 \][/tex]
[tex]\[ 108 \times 15 = 1620 \][/tex]
[tex]\[ 1620 \times 18 = 29160 \][/tex]
Thus,
[tex]\[ \log_5 (9 \times 12 \times 15 \times 18) = \log_5 29160 \][/tex]
3. Simplify the right-hand side:
Combine the logarithms on the right:
[tex]\[ 1 + \log_5 x + \log_5 x^2 = 1 + \log_5 (x \cdot x^2) = 1 + \log_5 x^3 \][/tex]
4. Convert 1 to a logarithm:
Since [tex]\(\log_5 5 = 1\)[/tex],
[tex]\[ 1 + \log_5 x^3 = \log_5 5 + \log_5 x^3 = \log_5 (5 \cdot x^3) \][/tex]
5. Set the logarithms equal to each other:
[tex]\[ \log_5 29160 = \log_5 (5x^3) \][/tex]
6. Remove the logarithms:
[tex]\[ 29160 = 5x^3 \][/tex]
7. Solve for [tex]\(x^3\)[/tex]:
[tex]\[ x^3 = \frac{29160}{5} = 5832 \][/tex]
8. Solve for [tex]\(x\)[/tex]:
Take the cube root of both sides:
[tex]\[ x = \sqrt[3]{5832} \][/tex]
Note that [tex]\(5832 = 18^3\)[/tex], so:
[tex]\[ x = 18 \][/tex]
Thus, the solution for the second equation is:
[tex]\[ x = 18 \][/tex]
### Summary
1. [tex]\( r = 2^{\frac{64}{3}} \)[/tex]
2. [tex]\( x = 18 \)[/tex]
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