Evaluate the following exponential functions: f(x) = 2^3 when x = 3, f(x) = 5^(x+1) when x = 2, f(x) = e^x when x = 1, f(x) = 2^(x-1) when x = 2. Convert the logarithmic form into... Evaluate the following exponential functions: f(x) = 2^3 when x = 3, f(x) = 5^(x+1) when x = 2, f(x) = e^x when x = 1, f(x) = 2^(x-1) when x = 2. Convert the logarithmic form into its equivalent logarithmic form.

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Understand the Problem

The question involves evaluating various exponential functions for specific values of x and converting logarithmic expressions into their equivalent forms. It seems to focus on properties of exponential and logarithmic functions.

Answer

1. $f(3) = 8$ 2. $f(2) = 125$ 3. $f(1) = e$ 4. $f(2) = 2$ 5. $f(2) = 1$ 6. $\log_2 8 = 3$, $\log_3 3 = 1$, $\log_4 16 = 2$
Answer for screen readers
  1. $f(3) = 8$
  2. $f(2) = 125$
  3. $f(1) = e$
  4. $f(2) = 2$
  5. $f(2) = 1$
  6. Logarithmic forms:
    • $\log_2 8 = 3$
    • $\log_3 3 = 1$
    • $\log_4 16 = 2$

Steps to Solve

  1. Evaluate $f(x) = 2^3$ when $x = 3$

Substituting $x = 3$ into the function gives: $$ f(3) = 2^3 = 8 $$

  1. Evaluate $f(x) = 5^{x+1}$ when $x = 2$

Substituting $x = 2$ into the function gives: $$ f(2) = 5^{2+1} = 5^3 = 125 $$

  1. Evaluate $f(x) = e^x$ when $x = 1$

Substituting $x = 1$ into the function: $$ f(1) = e^1 = e $$

  1. Evaluate $f(x) = 2^{x-1}$ when $x = 2$

Substituting $x = 2$ into the function gives: $$ f(2) = 2^{2-1} = 2^1 = 2 $$

  1. Evaluate $f(x) = 4^{2-x}$ when $x = 2$

Substituting $x = 2$ into the function gives: $$ f(2) = 4^{2-2} = 4^0 = 1 $$

  1. Convert the logarithmic expressions into equivalent forms
  • For $2^3 = 8$, the logarithmic form is: $$ \log_2 8 = 3 $$

  • For $3^1 = 3$, the logarithmic form is: $$ \log_3 3 = 1 $$

  • For $4^2 = 16$, the logarithmic form is: $$ \log_4 16 = 2 $$

  1. $f(3) = 8$
  2. $f(2) = 125$
  3. $f(1) = e$
  4. $f(2) = 2$
  5. $f(2) = 1$
  6. Logarithmic forms:
    • $\log_2 8 = 3$
    • $\log_3 3 = 1$
    • $\log_4 16 = 2$

More Information

The calculations show how to evaluate exponential functions and convert certain exponential equations into logarithmic form. The use of $e$ represents Euler's number, an important constant in mathematics.

Tips

  • Confusing the base and the exponent when substituting values in exponential functions.
  • Misapplying the properties of logarithms, specifically in finding the equivalent logarithmic forms.

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