What is x if ln x 0

Solve logarithmic equations

Equations that contain logarithms are Logarithmic equations. In the expression lieda(x) are a ≠ 1 and x> 0. Some logarithm equations can be solved using the logarithmic laws. As a rule, an expression that consists of several logarithms must be rewritten in such a way that only one logarithm occurs.

example 1

Solve the logarithm equation log10(4x + 5) = 2

log10(4x + 5)=2Expose with the base of the logarithm (here: 10)
10log10(4x + 5)=102Rewrite with the law of logarithms
4x + 5=102-5
4x=100 - 5÷4
x=$ \ frac {95} {4} $ = 23.75-5

Example 2

Solve the logarithm equation log10(x + 3) + log10(x) = 1

log10(x + 3) + log10(x)=1Contract logarithms
log10(x [x + 3])=1Expose with the base of the logarithm
10log10(x [x + 3])=101Rewrite with the law of logarithms
x (x + 3)=10Multiply out
x2 + 3x=10-10
x2 + 3x - 10=0Factoring
(x - 2) (x + 5)=0To solve
=>x1 = 2, x2 = -5

Although this equation has two solutions, the logarithm of a negative number is not defined, as is the case with x2 = -5 would be the case. We can check this simply by plugging it back into the original equation. Therefore, this logarithm equation only has the solution x1 = 2

Example 3

Solve the logarithm equation ln (x - 3) + ln (x - 2) = ln (12x + 24)

ln (x - 3) + ln (x - 2)=ln (12x + 24)Summarize the sum of two logarithms with the same base as the product of a logarithm
ln ((x - 3) (x - 2))=ln (12x + 24)Bring all logarithms to one page
ln ((x - 3) (x - 2)) - ln (12x + 24)=0Summarize the difference between two as the quotient of a logarithm
$ \ ln \ left ({\ dfrac {({x-3}) \ cdot ({x-2})} {12x + 24}} \ right) $=0Expose with the base (here e) and resolve the logarithms
$ {\ dfrac {({x-3}) \ cdot ({x-2})} {12x + 24}} $=1Multiply both sides by 12x + 24
(x - 3) (x - 2)=12x + 24Multiply the left side
x2 - 5x + 6=12x + 24Subtract 12x + 24 from both sides
x2 - 17x - 18=0Factor the equation
(x - 18) (x + 1)=0This allows us to split the equation into two new equations
x - 18 = 0 or x + 1 = 0Solve equations
x = 18 or x = -1Replace solutions in the equation and check
x = 18

By inserting it again, we find that there is only one solution, namely x = 18.