Question

Exponential equations examples

Answer 1

[AI-generated answer — Bot GPT 3.5]

1) Solve the exponential equation 2^x = 16.

To solve this equation, we need to find the value of x that makes 2^x equal to 16. We can rewrite 16 as a power of 2 by taking the base-2 logarithm of both sides:

log2(2^x) = log2(16)

x = log2(16)

Using the change of base formula, we can rewrite this as:

x = log(16) / log(2)

Simplifying further, we get:

x ≈ 4

Therefore, the solution to the exponential equation 2^x = 16 is x = 4.

2) Solve the exponential equation 3^(2x+1) = 27.

To solve this equation, we need to isolate the base 3 on one side of the equation. We can start by taking the base-3 logarithm of both sides:

log3(3^(2x+1)) = log3(27)

(2x+1) = log3(27)

Since 27 can be expressed as 3^3, we can simplify further:

(2x+1) = 3

Solving for x, we get:

2x = 2

x = 1

Therefore, the solution to the exponential equation 3^(2x+1) = 27 is x = 1.

3) Solve the exponential equation e^x = 10.

To solve this equation, we can take the natural logarithm (base-e logarithm) of both sides:

ln(e^x) = ln(10)

x = ln(10)

Using a calculator to approximate ln(10), we get:

x ≈ 2.3026

Therefore, the solution to the exponential equation e^x = 10 is x ≈ 2.3026.