To determine which equations require logarithms to solve for the given variable, we need to identify cases where the variable is in the exponent.

Let's analyze each option:

• (A) $A = 10,000 (1.01)^{13}$

  • The exponent is a constant (13), so no logarithms are needed.
  • Not a correct choice.

• (B) $25,000 = 10,000 (1.01)^{2t}$

  • The variable ($t$) is in the exponent.
  • To solve for $t$, we take the logarithm.
  • Correct choice.

• (C) $1,000 (0.55)^x = 4$

  • The variable ($x$) is in the exponent.
  • To solve for $x$, we take the logarithm.
  • Correct choice.

• (D) $1,000 (x)^4 = 14$

  • The variable ($x$) is the base, not the exponent.
  • Solve by taking the fourth root, not logarithms.
  • Not a correct choice.

• (E) $1,400 = \frac{2}{3} (1 + \frac{.05}{12})^{12t}$

  • The variable ($t$) is in the exponent.
  • To solve for $t$, we take the logarithm.
  • Correct choice.

Final Answer:

✔ (B), (C), and (E) require logarithms to solve.

Would you like a step-by-step solution for one of these?

Related Questions:

1. How do logarithms help solve exponential equations?
2. Can you solve $5^x = 20$ using logarithms?
3. What is the logarithmic form of $a^b = c$?
4. How do you use the natural logarithm (ln) to solve for exponents?
5. What is the difference between log and ln?

Tip:

To solve an equation of the form $a^x = b$, use logarithms:
$x = \frac{\log b}{\log a}$