# The Natural Base e

The natural base e, often denoted as "e," is a fundamental mathematical constant representing the base of the natural logarithm. It is approximately equal to 2.71828 and plays a crucial role in various branches of mathematics, particularly in calculus and exponential functions. Discovered by the Swiss mathematician Leonhard Euler, e arises naturally in many mathematical and scientific contexts, offering unique properties and applications. Its significance extends to compound interest calculations, continuous growth and decay models, and the study of rates of change in calculus, making it an indispensable constant in mathematical analysis and theoretical physics.

Questions

- The price of 9-volt batteries is increasing according to the function #P(t) = 1.1e^(0.047t)# , where #t # is years after January 1, 1980. During what year will the price reach $5?
- How do you find the inverse of #y = ln(x) + ln(x-6)#?
- How do you solve #e^(x+1) = 30#?
- Let # f(x) # be the function # f(x) = 5^x - 5^{-x}. # Is # f(x) # even, odd, or neither ? Prove your result.
- Use your calculator to evaluate e^-3. Round your answer to three decimal places.?
- How do you simplify #e^x*e^(-3x)*e^4#?
- How do you solve this using logarithms?
- How do you solve #10^ { x } \cdot e ^ { x } = 3#?
- How do you expand #log_6(x*y)#?
- How do you solve #e^ { y } = 6#?
- If #2^x + 2^x + 2^x + 2^x = 2^7#, what is the value of #x#?
- How do you simplify #1/2 (log_bM + log_bN - log_bP)#?
- Show that 1-:logx to the base 2 +1-:logx to the base 3 +........+1-:logx to the base 43 =1-:log x to the base 431 ?
- If f(a + b) = f(a) + f(b) - 2f(ab) for all nonnegative integers a and b, and f(1) = 1, compute f(1986). ???????
- How do you solve #3-4e^x=-1#?
- How do you simplify this? ln e ^78
- How do you simplify #(4e^x)/e^(4x)#?
- Is the number e rational or irrational?
- How do I add two numbers with the same base but different exponents?
- A bacteria population grows such that growth rate is proportional to population. At t=0 there are 100000 bacteria. At 48 hours there are 300000. How many bacteria will there be at t=72 hours?