IDNLearn.com makes it easy to find the right answers to your questions. Join our knowledgeable community and access a wealth of reliable answers to your most pressing questions.
Sagot :
Certainly! Let's solve the exponential equation [tex]\( 23^x = 6 \)[/tex] step by step.
1. Understanding the Equation:
The equation we need to solve is [tex]\( 23^x = 6 \)[/tex]. We need to find the value of [tex]\( x \)[/tex] that makes this equation true.
2. Applying the Logarithm:
To solve for [tex]\( x \)[/tex], we can take the natural logarithm (or common logarithm) of both sides of the equation. This is because taking the logarithm allows us to bring the exponent down as a multiplier, making it easier to solve for [tex]\( x \)[/tex].
[tex]\[ \log(23^x) = \log(6) \][/tex]
3. Using Logarithm Properties:
By using the power rule of logarithms, which states that [tex]\( \log(a^b) = b \log(a) \)[/tex], we can rewrite the left-hand side of the equation:
[tex]\[ x \cdot \log(23) = \log(6) \][/tex]
4. Solving for [tex]\( x \)[/tex]:
Now, we need to isolate [tex]\( x \)[/tex]. To do this, we divide both sides of the equation by [tex]\( \log(23) \)[/tex]:
[tex]\[ x = \frac{\log(6)}{\log(23)} \][/tex]
5. Calculating the Value:
Using a calculator to evaluate the right-hand side, we get:
[tex]\[ x \approx 0.5714440358797147 \][/tex]
Thus, the value of [tex]\( x \)[/tex] that satisfies the equation [tex]\( 23^x = 6 \)[/tex] is approximately [tex]\( 0.5714440358797147 \)[/tex].
By following these steps, we have determined that the solution to the equation [tex]\( 23^x = 6 \)[/tex] is [tex]\( x \approx 0.5714440358797147 \)[/tex].
1. Understanding the Equation:
The equation we need to solve is [tex]\( 23^x = 6 \)[/tex]. We need to find the value of [tex]\( x \)[/tex] that makes this equation true.
2. Applying the Logarithm:
To solve for [tex]\( x \)[/tex], we can take the natural logarithm (or common logarithm) of both sides of the equation. This is because taking the logarithm allows us to bring the exponent down as a multiplier, making it easier to solve for [tex]\( x \)[/tex].
[tex]\[ \log(23^x) = \log(6) \][/tex]
3. Using Logarithm Properties:
By using the power rule of logarithms, which states that [tex]\( \log(a^b) = b \log(a) \)[/tex], we can rewrite the left-hand side of the equation:
[tex]\[ x \cdot \log(23) = \log(6) \][/tex]
4. Solving for [tex]\( x \)[/tex]:
Now, we need to isolate [tex]\( x \)[/tex]. To do this, we divide both sides of the equation by [tex]\( \log(23) \)[/tex]:
[tex]\[ x = \frac{\log(6)}{\log(23)} \][/tex]
5. Calculating the Value:
Using a calculator to evaluate the right-hand side, we get:
[tex]\[ x \approx 0.5714440358797147 \][/tex]
Thus, the value of [tex]\( x \)[/tex] that satisfies the equation [tex]\( 23^x = 6 \)[/tex] is approximately [tex]\( 0.5714440358797147 \)[/tex].
By following these steps, we have determined that the solution to the equation [tex]\( 23^x = 6 \)[/tex] is [tex]\( x \approx 0.5714440358797147 \)[/tex].
We value your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. For precise answers, trust IDNLearn.com. Thank you for visiting, and we look forward to helping you again soon.