To solve the exponential equation $ 2^{x} = 16 $, follow these steps:

### Step 1: Express Both Sides with the Same Base

First, notice that 16 can be expressed as a power of 2:
\[
16 = 2^{4}
\]
So, the equation becomes:
\[
2^{x} = 2^{4}
\]

### Step 2: Set the Exponents Equal

Since the bases are the same and the equation $ 2^{x} = 2^{4} $ is true, the exponents must be equal:
\[
x = 4
\]

### Alternative Method: Using Logarithms

If you're more comfortable using logarithms, here's how you can solve the equation:

1. **Take the logarithm of both sides.** You can use any logarithm base, but common logarithm (base 10) or natural logarithm (base $ e $) are typically used.

\[
\log(2^{x}) = \log(16)
\]

2. **Apply the power rule of logarithms**, which states that $ \log(a^{b}) = b \cdot \log(a) $:

\[
x \cdot \log(2) = \log(16)
\]

3. **Solve for $ x $**:

\[
x = \frac{\log(16)}{\log(2)}
\]

4. **Calculate the values** (using base 10 logarithms):

\[
x = \frac{\log(16)}{\log(2)} = \frac{1.2041}{0.3010} \approx 4
\]

Thus, the solution is:

\[
\boxed{4}
\]

Home Question Bank Math Algebra

Question

Solve the exponential equation: $ 2^{x} = 16 $

Ask by Ortiz Romero.

Dec 23,2024 22:08

UpStudy AI Solution

Tutor-Verified Answer

Answer

$ x = 4 $

Solution

To solve the exponential equation $ 2^{x} = 16 $, follow these steps: ### Step 1: Express Both Sides with the Same Base First, notice that 16 can be expressed as a power of 2: \[ 16 = 2^{4} \] So, the equation becomes: \[ 2^{x} = 2^{4} \] ### Step 2: Set the Exponents Equal Since the bases are the same and the equation $ 2^{x} = 2^{4} $ is true, the exponents must be equal: \[ x = 4 \] ### Alternative Method: Using Logarithms If you're more comfortable using logarithms, here's how you can solve the equation: 1. **Take the logarithm of both sides.** You can use any logarithm base, but common logarithm (base 10) or natural logarithm (base $ e $) are typically used. \[ \log(2^{x}) = \log(16) \] 2. **Apply the power rule of logarithms**, which states that $ \log(a^{b}) = b \cdot \log(a) $: \[ x \cdot \log(2) = \log(16) \] 3. **Solve for $ x $**: \[ x = \frac{\log(16)}{\log(2)} \] 4. **Calculate the values** (using base 10 logarithms): \[ x = \frac{\log(16)}{\log(2)} = \frac{1.2041}{0.3010} \approx 4 \] Thus, the solution is: \[ \boxed{4} \]

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Beyond the Answer

To solve the equation $ 2^{x} = 16 $, first, we can express 16 as a power of 2. Since $ 16 = 2^4 $, we can rewrite the equation as $ 2^{x} = 2^{4} $. When the bases are the same, you can set the exponents equal to each other, leading to $ x = 4 $. So, the solution to the equation is $ x = 4 $. Now, here's a little twist: did you know that exponential equations like this one pop up in real life every day? From calculating compound interest to predicting population growth, understanding how to solve them is a total game-changer! You’ll find they’re especially handy in finance and science, where growth and decay can be modeled with similar equations. Also, a common mistake when solving exponential equations is forgetting that the bases must be the same to use the property of exponents. Always double-check how expressions can be rewritten. It's like trying to open a locked door; sometimes you just need the right key (or in this case, the right base) to get in!