Given that [tex]e^x=120[/tex], write this as a logarithm and find the value.
Answer (2)
Rewrite the exponential equation e x = 120 in logarithmic form: x = ln ( 120 ) .
Calculate the value of ln ( 120 ) using a calculator: ln ( 120 ) ≈ 4.78749 .
The value of x is approximately 4.78749.
Therefore, the final answer is 4.78749 .
Explanation
Understanding the Problem We are given the equation e x = 120 and asked to express it as a logarithm and find the value of x .
Converting to Logarithmic Form To rewrite the exponential equation e x = 120 in logarithmic form, we use the definition of the natural logarithm. The natural logarithm, denoted as ln , is the logarithm to the base e . Therefore, if e x = y , then x = ln ( y ) .
Expressing as a Logarithm Applying this to our equation e x = 120 , we get x = ln ( 120 ) .
Calculating the Value Now, we need to find the value of ln ( 120 ) . Using a calculator, we find that ln ( 120 ) ≈ 4.78749 .
Examples
Logarithms are incredibly useful in many real-world scenarios, especially when dealing with exponential growth or decay. For example, calculating the time it takes for an investment to double at a certain interest rate involves logarithms. If you invest P dollars at an annual interest rate r compounded continuously, the amount A after t years is given by A = P e r t . To find out how long it takes for your investment to double (i.e., A = 2 P ), you would solve for t using logarithms: 2 P = P e r t ⇒ 2 = e r t ⇒ ln ( 2 ) = r t ⇒ t = r l n ( 2 ) . This shows how logarithms help us solve for exponents in practical financial problems.
The equation e x = 120 can be rewritten in logarithmic form as x = ln ( 120 ) . Calculating this gives a value of approximately 4.78749 for x . Thus, the solution is x ≈ 4.78749 .
;
