## Are logarithmic functions inverses?

The logarithmic function g(x) = logb(x) is the inverse of the exponential function f(x) = bx. The meaning of y = logb(x) is by = x.

## Is logarithmic function differentiable?

Theorem. The (real) natural logarithm function is differentiable.

**Why the exponential function is known as the inverse of logarithmic function?**

If the logarithm is understood as the inverse of the exponential function, then the properties of logarithms will naturally follow from our understanding of exponents. The meaning of the logarithm. The logarithmic function g(x) = logb(x) is the inverse of the exponential function f(x) = bx.

**Is logarithmic function differentiable everywhere?**

Theorem 8.1 log x is defined for all x > 0. It is everywhere differentiable, hence continuous, and is a 1-1 function. The Range of log x is (−∞, ∞).

### Is log the same as ln?

The difference between log and ln is that log is defined for base 10 and ln is denoted for base e. For example, log of base 2 is represented as log2 and log of base e, i.e. loge = ln (natural log).

### What is the relationship between logarithms and exponentials?

Logarithms are the “opposite” of exponentials, just as subtraction is the opposite of addition and division is the opposite of multiplication. Logs “undo” exponentials. Technically speaking, logs are the inverses of exponentials.

**How can the inverse relationship between an exponential function and its inverse logarithmic function be explained?**

Logarithmic functions are the inverses of exponential functions. The inverse of the exponential function y = ax is x = ay. The logarithmic function y = logax is defined to be equivalent to the exponential equation x = ay.

**What must be true about a function for its inverse to also be a function?**

If the function has an inverse that is also a function, then there can only be one y for every x. A one-to-one function, is a function in which for every x there is exactly one y and for every y, there is exactly one x. A one-to-one function has an inverse that is also a function.