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Sagot :
Certainly! Let's solve for [tex]\( x \)[/tex] in the equation [tex]\( \log_9(x) = -2 \)[/tex].
### Step-by-Step Solution:
1. Understand the Logarithmic Equation:
The given equation is [tex]\( \log_9(x) = -2 \)[/tex]. This is a logarithmic equation where the base of the logarithm is 9 and the result is [tex]\(-2\)[/tex].
2. Rewrite the Equation Using Exponential Form:
By the definition of logarithms, if [tex]\( \log_b(a) = c \)[/tex], then [tex]\( a = b^c \)[/tex].
In our problem:
[tex]\[ \log_9(x) = -2 \][/tex]
This implies:
[tex]\[ x = 9^{-2} \][/tex]
3. Evaluate the Exponential Expression:
Now, evaluate what [tex]\( 9^{-2} \)[/tex] means.
[tex]\[ 9^{-2} = \frac{1}{9^2} \][/tex]
[tex]\[ 9^2 = 81 \][/tex]
So:
[tex]\[ 9^{-2} = \frac{1}{81} \][/tex]
4. Write the Final Answer:
Therefore, the value of [tex]\( x \)[/tex] is:
[tex]\[ x = \frac{1}{81} \][/tex]
When we convert the fraction [tex]\( \frac{1}{81} \)[/tex] to its decimal form, it results in approximately:
[tex]\[ x \approx 0.012345679012345678 \][/tex]
So, the simplified answer to the given logarithmic equation [tex]\( \log_9(x) = -2 \)[/tex] is:
[tex]\[ x = \frac{1}{81} \][/tex]
And in decimal form, it is approximately:
[tex]\[ x \approx 0.012345679012345678 \][/tex]
### Step-by-Step Solution:
1. Understand the Logarithmic Equation:
The given equation is [tex]\( \log_9(x) = -2 \)[/tex]. This is a logarithmic equation where the base of the logarithm is 9 and the result is [tex]\(-2\)[/tex].
2. Rewrite the Equation Using Exponential Form:
By the definition of logarithms, if [tex]\( \log_b(a) = c \)[/tex], then [tex]\( a = b^c \)[/tex].
In our problem:
[tex]\[ \log_9(x) = -2 \][/tex]
This implies:
[tex]\[ x = 9^{-2} \][/tex]
3. Evaluate the Exponential Expression:
Now, evaluate what [tex]\( 9^{-2} \)[/tex] means.
[tex]\[ 9^{-2} = \frac{1}{9^2} \][/tex]
[tex]\[ 9^2 = 81 \][/tex]
So:
[tex]\[ 9^{-2} = \frac{1}{81} \][/tex]
4. Write the Final Answer:
Therefore, the value of [tex]\( x \)[/tex] is:
[tex]\[ x = \frac{1}{81} \][/tex]
When we convert the fraction [tex]\( \frac{1}{81} \)[/tex] to its decimal form, it results in approximately:
[tex]\[ x \approx 0.012345679012345678 \][/tex]
So, the simplified answer to the given logarithmic equation [tex]\( \log_9(x) = -2 \)[/tex] is:
[tex]\[ x = \frac{1}{81} \][/tex]
And in decimal form, it is approximately:
[tex]\[ x \approx 0.012345679012345678 \][/tex]
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