Question

lesson 26 - solving logarithmic equations
score: 15/100 answered: 3/20
question 4
solve.
\\(\log_{9}(z) = -5\\)
\\(z = \square\\)
question help: video message instructor

Explanation:

Step1: Recall logarithm definition

The logarithmic equation $\log_{b}(a) = c$ is equivalent to $b^c = a$ (by the definition of a logarithm, where $b>0, b
eq1$).

Step2: Apply the definition to the given equation

For the equation $\log_{9}(z) = -5$, we have $b = 9$, $c = -5$, and $a = z$. Using the equivalence from Step 1, we rewrite it as $z = 9^{-5}$.

Step3: Simplify the expression for $z$

We know that $9 = 3^2$, so $9^{-5}=(3^2)^{-5}$. Using the exponent rule $(a^m)^n=a^{m\times n}$, we get $(3^2)^{-5}=3^{2\times(-5)} = 3^{-10}$. Also, $a^{-n}=\frac{1}{a^n}$, so $3^{-10}=\frac{1}{3^{10}}$. Calculating $3^{10}=59049$, so $z=\frac{1}{59049}$.

Answer:

$\frac{1}{59049}$