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**Exponential and Logarithmic Functions**

The value of y = f_{n}(x) increases as n increases for n = 1,2,3....

y = f_{n}(x) = x^{n} increases as n increases âˆ€ n âˆˆ R^{+}

The growth of a polynomial function depends on the degree of the polynomial.

The higher the degree, the greater is the growth of the polynomial function.

The increase in f(x) = n^{x} is larger than the increase in f_{n}(x) = x^{n} âˆ€ n âˆˆ I^{+}, x âˆˆ R^{+}

An exponential function with a positive base b > 1 is a function defined by y = f(x) = b^{x}.

Exponential functions with base 10 are called common exponential functions.

For example, f(x) = 10^{x}.

Exponential functions with base e are called natural exponential functions.

For example, f(x) = e^{x} Where e = 1 + 1/1! + 1/2! + 1/3! + ....

**Features of exponential functions:**

The domain of exponential functions is R.

The range of exponential functions is the set of positive real numbers.

Graphs of exponential functions always grow upwards as we move from left to right on the X-axis.

Graphs of exponential functions are close to the X-axis, but never touch it for any large negative values of x.

Point (0, 1) is always on the graph of any exponential function (since b^{o} = 1, for any real number b > 1.).

An exponential equation b^{x} = a, b > 1 and b âˆˆ R then

log_{b}a = x.

log_{b} : R^{+} â†’ R x â†’ log_{b}x = y if b^{y} = x.

Note:

Logarithm of only positive real numbers can be found and its value can be any real number.

Logarithms to base 10 are called common logarithms.

For example: log_{10} 425

Logarithms to base e are called natural logarithms.

For example: log_{e} 23

**Important facts about logarithmic functions:**

The logarithm of a non-positive number cannot be defined. Hence, the domain of a logarithmic function is R^{+}.

The range of a logarithmic function is the set of all real numbers.

Point (0, 1) is always on the graph of any logarithmic function.

Graphs of logarithmic functions grow as we move from left to right on the X-axis.

Graphs of logarithmic functions are close to the Y-axis, but never touch it for any small positive values of x.

Graphs of y = e^{x} and log x are mirror images of each other reflected in the line, y = x.

**Change of base rule**:

log_{a}p = log_{b}p/log_{b}a

Let log_{a}p = Î±

log_{b}p = ÃŸ

log_{b}a = Î“

a^{Î±} = p ......(1), b^{ÃŸ} = p ......(2) , b^{Î“} = a .....(3)

From (1) and (3), p = a^{Î±} = (b^{Î“})^{Î±} = b^{Î“Î±} ....(4)

From (2) and (4), b^{ÃŸ} = p = b^{Î“Î±}

â‡’ ÃŸ = Î“Î± â‡’ Î± = ÃŸ/Î“

âˆ´ log_{a}p = log_{b}p/log_{b}a

The logarithm of the product of two positive numbers to any base is equal to the sum of the logarithms of the numbers to the same bases.

log_{b}pq = log_{b}p + log_{b}q

Let log_{b}pq = Î±

log_{b}p = ÃŸ, log_{b}q = Î“

b^{Î±} = pq ....(1), b^{ÃŸ} = p .....(2), b^{Î“} = q .....(3)

From (1), (2) and (3), b^{Î±} = pq = b^{ÃŸ}b^{Î“}

â‡’ b^{Î±} = b^{(ÃŸ+Î“) } â‡’ Î± = ÃŸ + Î“

âˆ´ log_{b}pq = log_{b}p + log_{b}q

Note :

Suppose p = q â‡’ log_{b}p^{2} = 2 log_{b}p

This can be generalised log_{b}p^{n} = n log_{b}p, where n is a real number.

Similarly logarithm of the quotient of two numbers to any base is equal to the logarithm of the numerator minus the logarithm of the denominator to the same base i.e.

log_{b} (p/q) = log_{b} p - log_{b} q

**Exponential and Logarithmic Functions**

The value of y = f_{n}(x) increases as n increases for n = 1,2,3....

y = f_{n}(x) = x^{n} increases as n increases âˆ€ n âˆˆ R^{+}

The growth of a polynomial function depends on the degree of the polynomial.

The higher the degree, the greater is the growth of the polynomial function.

The increase in f(x) = n^{x} is larger than the increase in f_{n}(x) = x^{n} âˆ€ n âˆˆ I^{+}, x âˆˆ R^{+}

An exponential function with a positive base b > 1 is a function defined by y = f(x) = b^{x}.

Exponential functions with base 10 are called common exponential functions.

For example, f(x) = 10^{x}.

Exponential functions with base e are called natural exponential functions.

For example, f(x) = e^{x} Where e = 1 + 1/1! + 1/2! + 1/3! + ....

**Features of exponential functions:**

The domain of exponential functions is R.

The range of exponential functions is the set of positive real numbers.

Graphs of exponential functions always grow upwards as we move from left to right on the X-axis.

Graphs of exponential functions are close to the X-axis, but never touch it for any large negative values of x.

Point (0, 1) is always on the graph of any exponential function (since b^{o} = 1, for any real number b > 1.).

An exponential equation b^{x} = a, b > 1 and b âˆˆ R then

log_{b}a = x.

log_{b} : R^{+} â†’ R x â†’ log_{b}x = y if b^{y} = x.

Note:

Logarithm of only positive real numbers can be found and its value can be any real number.

Logarithms to base 10 are called common logarithms.

For example: log_{10} 425

Logarithms to base e are called natural logarithms.

For example: log_{e} 23

**Important facts about logarithmic functions:**

The logarithm of a non-positive number cannot be defined. Hence, the domain of a logarithmic function is R^{+}.

The range of a logarithmic function is the set of all real numbers.

Point (0, 1) is always on the graph of any logarithmic function.

Graphs of logarithmic functions grow as we move from left to right on the X-axis.

Graphs of logarithmic functions are close to the Y-axis, but never touch it for any small positive values of x.

Graphs of y = e^{x} and log x are mirror images of each other reflected in the line, y = x.

**Change of base rule**:

log_{a}p = log_{b}p/log_{b}a

Let log_{a}p = Î±

log_{b}p = ÃŸ

log_{b}a = Î“

a^{Î±} = p ......(1), b^{ÃŸ} = p ......(2) , b^{Î“} = a .....(3)

From (1) and (3), p = a^{Î±} = (b^{Î“})^{Î±} = b^{Î“Î±} ....(4)

From (2) and (4), b^{ÃŸ} = p = b^{Î“Î±}

â‡’ ÃŸ = Î“Î± â‡’ Î± = ÃŸ/Î“

âˆ´ log_{a}p = log_{b}p/log_{b}a

The logarithm of the product of two positive numbers to any base is equal to the sum of the logarithms of the numbers to the same bases.

log_{b}pq = log_{b}p + log_{b}q

Let log_{b}pq = Î±

log_{b}p = ÃŸ, log_{b}q = Î“

b^{Î±} = pq ....(1), b^{ÃŸ} = p .....(2), b^{Î“} = q .....(3)

From (1), (2) and (3), b^{Î±} = pq = b^{ÃŸ}b^{Î“}

â‡’ b^{Î±} = b^{(ÃŸ+Î“) } â‡’ Î± = ÃŸ + Î“

âˆ´ log_{b}pq = log_{b}p + log_{b}q

Note :

Suppose p = q â‡’ log_{b}p^{2} = 2 log_{b}p

This can be generalised log_{b}p^{n} = n log_{b}p, where n is a real number.

Similarly logarithm of the quotient of two numbers to any base is equal to the logarithm of the numerator minus the logarithm of the denominator to the same base i.e.

log_{b} (p/q) = log_{b} p - log_{b} q