Let's look at the problem in the image you uploaded:

The question asks us to solve the following equations and round the answers to the hundredth decimal place:

a. $5^x = 14$

b. $\log_{}(x + 3x + 9) = 2$

c. $10^x = 240$

d. $\ln x = 7$

Let's start solving each one.

a) Solve $5^x = 14$

To solve for $x$, take the logarithm of both sides: $x \log 5 = \log 14$ $x = \frac{\log 14}{\log 5}$ Using logarithmic values: $\log 14 \approx 1.1461 \quad \text{and} \quad \log 5 \approx 0.69897$ $x = \frac{1.1461}{0.69897} \approx 1.639$ So, $x \approx 1.64$ (rounded to the hundredth place).

b) Solve $\log_{}(x + 3x + 9) = 2$

Simplify the expression inside the logarithm: $\log_{}(4x + 9) = 2$ To solve for $x$, rewrite the logarithmic equation in its exponential form: $4x + 9 = 10^2$ $4x + 9 = 100$ Now, subtract 9 from both sides: $4x = 91$ $x = \frac{91}{4} = 22.75$ So, $x = 22.75$.

c) Solve $10^x = 240$

To solve for $x$, take the logarithm of both sides: $x \log 10 = \log 240$ Since $\log 10 = 1$, the equation simplifies to: $x = \log 240$ Using the logarithmic value: $\log 240 \approx 2.3802$ So, $x \approx 2.38$.

d) Solve $\ln x = 7$

To solve for $x$, rewrite the equation in its exponential form: $x = e^7$ Using the value of $e^7$: $e^7 \approx 1096.633$ So, $x \approx 1096.63$.

Summary of solutions:

• a) $x \approx 1.64$
• b) $x = 22.75$
• c) $x \approx 2.38$
• d) $x \approx 1096.63$

Would you like further details on any of these steps or need more questions answered? Here are some related questions that may help:

1. How do logarithms and exponents relate to each other?
2. What are some properties of logarithms to make solving them easier?
3. How can we apply natural logarithms in real-world problems?
4. What is the difference between logarithms with different bases?
5. How do you handle logarithmic equations with multiple terms inside the log?

Tip: When working with logarithms, remember that logarithms of numbers between 0 and 1 will be negative!