Part 1: Express in Exponential Form

To convert from logarithmic form $\ln(x) = y$ to exponential form, use the relationship:

$\ln(a) = b \implies e^b = a$

(a) $\ln(5) = 2y$

Convert to exponential form: $e^{2y} = 5$

(b) $\ln(t + 9) = -1$

Convert to exponential form: $e^{-1} = t + 9$

Simplify further if needed: $t + 9 = \frac{1}{e}$

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Part 2: Express in Logarithmic Form

To convert from exponential form $a^b = c$ to logarithmic form, use the relationship:

$a^b = c \implies \log_a(c) = b$

(a) $4^{-1/2} = 0.5$

Convert to logarithmic form: $\log_4(0.5) = -\frac{1}{2}$

(b) $\frac{1}{3}^{-3} = 27$

Simplify $\frac{1}{3}^{-3}$ first: $\frac{1}{3}^{-3} = \left( \frac{1}{3} \right)^{-3} = 3^3 = 27$ Now, convert to logarithmic form: $\log_{\frac{1}{3}}(27) = -3$

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Would you like additional details or explanations on these conversions?

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Related Questions

1. What is the general relationship between logarithms and exponents?
2. How does the base of a logarithm affect its exponential equivalent?
3. Can you solve $\ln(x) = 3$ in exponential form and find $x$?
4. What is the value of $\ln(e)$, and why is it significant in logarithmic operations?
5. How would you solve $\log_2(x) = 5$ in exponential form?

Tip

Always remember: the natural logarithm $\ln(x)$ uses $e$ as its base, where $e \approx 2.718$.