Home >> Pre-Calculus >> 3. Exponential and Logarithmic Functions

<< 3.3. Properties of Logarithms

**>> 3.5. Modeling with Exponential and Logarithmic Functions**

3.4. Exponential and Logarithmic Equations

We are often presented with equations involving exponential or logarithmic functions. In this section we show how to solve the simplest of these equations, i.e. equations of one of the two following forms:

**(3.4.1)**b^{x}= c

**(3.4.2)**log_{b}x = c

Both of these equations are easy to solve using Equations (3.3.6) and (3.3.7) respectively. For instance, taking the base-b logarithm of both sides of (3.4.1) and using (3.3.5) with b=10, we obtain

log_{b}(b^{x}) = log_{b}c = log c / log b.

But by (3.3.6), we see that the left side is just x. Thus we obtain the following solution to (3.4.1):

**(3.4.3)**x = log_{b}c.

Exponentiating both sides of (3.4.2) with base b, we obtain

Using (3.3.7), we see once again that the left side is x, whence we have the solution

**(3.4.4)**x = b^{c}.

**Example 1:** Solve the exponential equation 2^{x} = 10.

**Solution: **The solution is given by x = log_{2}10 = log 10 / log 2 = 1 / log 2, which one easily calculates to be approximately 3.32 using a scientific calculator.

**Example 2:** Solve the exponential equation 0.5^{x} = 0.04.

**Solution: ** The solution is given by x = log_{0.5}0.04 = log 0.04 / log 0.5 = 4.64.

**Example 3:** Solve the logarithmic equation log x = 4.26.

**Solution: ** The solution is given by x = 10^{4.26} ≈ 18,197.

**Example 4:** Solve the logarithmic equation log_{π}x = 1.76.

**Solution: ** The solution is given by x = π^{1.76} ≈ 7.50.

Home >> Pre-Calculus >> 3. Exponential and Logarithmic Functions