Exponential equations examples
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.