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2021-08-02

Exponential and Logarithmic Equations solve the equation. Find the exact solution if possible. otherwise, use a calculator to approximate to two decimals.

${\mathrm{log}}_{x}(x+5)-{\mathrm{log}}_{8}(x-2)=1$

Macsen Nixon

Skilled2021-08-03Added 117 answers

Step 1

Law of logarithm:

Consider m to be a positive number and$m\ne q1.$

Again consider M and M to be any real numbers with$M>0$ and $N>0.$

The difference of logarithms of two numbers is equal to the logarithm of quotient of two numbers as,

${\mathrm{log}}_{m}M-{\mathrm{log}}_{m}N={\mathrm{log}}_{m}\left(\frac{M}{N}\right)$

The logarithm function with base m is denoted by$\mathrm{log}}_{m$ can be defined as,

${\mathrm{log}}_{m}M=y$

$M={m}^{y}$

Step 2

The given logarithm equation is,

1)${\mathrm{log}}_{8}(x+5)-{\mathrm{log}}_{8}(x-2)=1$

The above logarithm equation can be combined from the laws of logarithm as,

2)${\mathrm{log}}_{8}\frac{(x+5)}{(x-2)}=1$

The equation (2) can be expressed as,

$\frac{(x+5)}{(x-2)}={8}^{1}$

$x+5=8(x-2)$

$x+5=8x-16$

$x-8x=-16-5$

Simplify above equation as,

$-7x=-21$

$7x=21$

$x=\frac{21}{7}$

$x=3$

Therefore,$x=3$ is the solution of equation ${\mathrm{log}}_{8}(x+5)-{\mathrm{log}}_{8}(x-2)=1.$

Conclusion:

Thus, the solution of the logarithm equation${\mathrm{log}}_{8}(x+5)-{\mathrm{log}}_{8}(x-2)=1$ is 3.

Law of logarithm:

Consider m to be a positive number and

Again consider M and M to be any real numbers with

The difference of logarithms of two numbers is equal to the logarithm of quotient of two numbers as,

The logarithm function with base m is denoted by

Step 2

The given logarithm equation is,

1)

The above logarithm equation can be combined from the laws of logarithm as,

2)

The equation (2) can be expressed as,

Simplify above equation as,

Therefore,

Conclusion:

Thus, the solution of the logarithm equation