### 12.7 - Logarithmic equations

Before reading this section you may want to review the following topics:
A **logarithmic equation** is one where the variable to be solved for (call it *x*)
is in the argument of a logarithm function.
For example log(*x*) = 5 is a logarithmic equation while log(5) = *x* is not.
To solve a logarithmic equation follow these steps:

- If the equation contains several logarithms then you must first use the
properties of logarithms
to combine them into a single logarithm.

- Isolate the logarithm. This means put the equation into the form
log
_{ b}( *f* ) = *a* so that the log
function is alone on one side of the equation. (The expression
*f* contains the unknown *x*.)

- Antilog both sides, thus putting the
equation into the exponential form
*b*^{ a} = f.

- The unknown
*x* is no longer inside a logarithm.
Now you can finish solving for *x*
by using the basic procedures for solving equations.

- Check the solution.

**Example:** Solve the logarithmic equation
2 log_{ 3 }(*x* − 1) = 4 for *x*.

**Solution:** There is only a single logarithm so the first step is to isolate the
logarithm. To do this divide both sides by 2:
log_{ 3 }(*x* − 1) = 2

The next step is to antilog both sides:
*x* − 1 = 3^{ 2}

The equation is no longer logarithmic and we can finish solving for *x*
by simply adding 1 to both sides:
*x* = 10

**Example:** Solve the logarithmic equation
log (3 *x* + 1) − 2 log (*x*) = 1 for *x*:

**Solution:** The first step is to combine the logarithms using the
properties of logarithms.
First use property 3, then property 2:

log(3 *x* + 1) − log (*x*^{2 }) = 1

The next step is to antilog both sides.
Note that the base of the logarithm is understood to be 10.

The equation is no longer logarithmic - it is fractional, so we can proceed to solve for *x*
using techniques for fractional equations. Clear the denominator
by multiplying through by *x*^{2 } and then move all terms to the
left side:
10 *x*^{ 2} − 3 *x* − 1 = 0

The result is a quadratic equation in standard form.
The left-hand-side can be factored:

(5 *x* + 1) (2 *x* − 1) = 0.

We can replace this equation by two new equations, each of which results from setting
one of the factors equal to zero. Solving them yields the solutions:
Now we must check the solutions. Substituting *x* = 1/2 into the original
equation and simplifying yields the equation 1 = 1, so it checks out. But substituting
x = −1/5 into the original equation means that we must evaluate the logarithm of
a negative number and this cannot be done over the real numbers. Thus this solution
is extraneous; which leaves us with the only solution, *x* = 1/2.

Note that not all logarithmic equations can be solved using algebra.
For example consider the seemingly simple equation *x* = log (*x*).
We cannot get the *x* out of the logarithm without putting the other *x*
into an exponential. This equation can only be solved approximately using a computer.

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