Answer: z = 1
Step-by-step explanation:
We will solve the given equation for z.
Given:
[tex]\displaystyle log_{10}z+log_{10}(z+9)=1[/tex]
Add the logarithms together:
➜ [tex]loga + logb=log(ab)[/tex]
[tex]\displaystyle log_{10}(z*(z+9))=1[/tex]
Multiply:
[tex]\displaystyle log_{10}(z^{2} +9z)=1[/tex]
Exponential form:
[tex]10^1=z^2+9z[/tex]
To the power of 1:
[tex]10=z^2+9z[/tex]
Subtract 10 from both sides of the equation:
[tex]0=z^2+9z-10[/tex]
Factor:
➜ -1 and 10 add to 9 and multiply to -10.
[tex]0=(z-1)(z+10)[/tex]
Zero product property and separate the cases:
0 = z - 1 0 = z + 10
z = 1 z = -10
You cannot take the logarithm of a negative number (-10+9=-1), so our answer is 1.
