An equation in which the unknown is in the logarithm or in the logarithmic basis of at least one logarithm is called a **logarithmic equation.**

There is no general method for solving a logarithmic equation.

**Solving some types of logarithmic equations**

1) Solve equations of the type l** ogₐ x = b (**a, b ∈ R).

If** a> 0 **an

**1, then for any real number b,**

*d a ≠***the equation has a unique solution:**

2) Solving an equation of type (1)

The solutions of this equation are obtained by solving the equation:

**f (x) = g (x**) (2)

Every solution of equation (1) is also a solution of equation (2), but the opposite is not true, so the solutions of equation (2) will be checked in equation (1).

**Example 1.** Let us solve the logarithmic equation

**log₂ (x + 2) + log₂ (x + 14) = 6**.

This equation is equivalent to the equation:

*log₂ (x + 2) (х + 14) = log₂64*

The solutions of the equation

*(x + 2) (x + 14) = 64*

are: * х₁ = -18 *and

*2.*

**х₂ =****Check.*** For x = -18, the solved equation passes into the equation*

*log₂ (-16) + log₂ (-4) = 6*

which is untrue (logarithm cannot be a negative number). It follows that** x = -18 i**s not a solution of the equation under consideration.

For the solu** tion x** = 2, the solved equation becomes an exact equation, ie.

*log₂4 + log₂16 = 6, 2 + 4 = 6*

So, x =** 2 is **a solution of the equation.

3) Solve an equation of type* F (logₐ f (x)) = 0*

In the equati*on F (logₐ f (x)) = 0*

** a> 0 **and

**, an**

*a ≠ 1***x) is a given function o**

*d f (***x. With the ch**

*f**t it is reduced to the type: F*

**ange logₐ f (x) =**

**(t) = 0**If the equati** on F **(t) = 0 has soluti

**… tk, then the solution of the given equation comes down to solving the set of logarithmic equations**

*ons t₁, t₂, .*

**
**4) When solving some logarithmic equations, it is necessary to apply some of the known logarithmic identities.

**Example 3.** The equation

can be written like this:

and applying identity

the equation is obtained

that is