Can someone explain to me why ln e^5 equals 5?
ln 2.7185 = ln 148.413 = 5
ln (e^5 )= 5 ln e
but ln e = 1
so ln (e^5) = 5*1 = 5
How do you solve 4 to the power of 2-6x = 812? I can't find a common number that raised to a power equals 812.
Certainly! To understand why ln(e^5) equals 5, we need to understand what the natural logarithm (ln) and the mathematical constant e represent.
The natural logarithm (ln) is the logarithm to the base e. It is the inverse function of the exponential function, which means ln undoes the effect of the exponential function.
The mathematical constant e is an irrational number approximately equal to 2.71828. It appears frequently in various mathematical contexts, including growth and decay problems, compound interest, and continuous processes.
Now, to find the value of ln(e^5), we can break it down step by step.
Step 1: We have e^5 inside the ln function. The exponential function e^x represents the value of e raised to the power of x.
Step 2: Since e^5 is the argument of the ln function, it means we need to find the natural logarithm of e^5.
Step 3: Since the natural logarithm is the inverse function of the exponential function, the ln undoes the effect of e^5.
Step 4: The inverse relationship of ln and e^x implies that ln(e^x) will always equal x. In other words, the natural logarithm of e raised to any power x is always equal to x.
Therefore, ln(e^5) simplifies to 5, because the ln function undoes the effect of raising e to the power of 5, leaving us with the original power, which is 5.