Evaluate: a) $$ \log _{19}(\sqrt{19})=\square $$ b) $$ \log _{4}\left(\frac{1}{16}\right)=\square $$ c) $$ \log _{8}(64)=\square $$

Question

UpStudy AI · Accepted Answer

To evaluate the logarithmic expressions, we can use the properties of logarithms.

a) $$ \log_{19}(\sqrt{19}) $$

We can rewrite $$ \sqrt{19} $$ as $$ 19^{1/2} $$. Therefore, we have:

$$
\log_{19}(\sqrt{19}) = \log_{19}(19^{1/2})
$$

Using the property of logarithms $$ \log_b(a^c) = c \cdot \log_b(a) $$, we get:

$$
\log_{19}(19^{1/2}) = \frac{1}{2} \cdot \log_{19}(19)
$$

Since $$ \log_{19}(19) = 1 $$, we have:

$$
\log_{19}(\sqrt{19}) = \frac{1}{2} \cdot 1 = \frac{1}{2}
$$

So, the answer is:

$$
\log_{19}(\sqrt{19}) = \frac{1}{2}
$$

b) $$ \log_{4}\left(\frac{1}{16}\right) $$

We can rewrite $$ \frac{1}{16} $$ as $$ 16^{-1} $$ and also note that $$ 16 = 4^2 $$. Thus, we have:

$$
\frac{1}{16} = 16^{-1} = (4^2)^{-1} = 4^{-2}
$$

Now we can write:

$$
\log_{4}\left(\frac{1}{16}\right) = \log_{4}(4^{-2})
$$

Using the property of logarithms again:

$$
\log_{4}(4^{-2}) = -2 \cdot \log_{4}(4)
$$

Since $$ \log_{4}(4) = 1 $$, we have:

$$
\log_{4}\left(\frac{1}{16}\right) = -2 \cdot 1 = -2
$$

So, the answer is:

$$
\log_{4}\left(\frac{1}{16}\right) = -2
$$

c) $$ \log_{8}(64) $$

We can rewrite $$ 64 $$ as $$ 8^{2} $$ because $$ 8 = 2^3 $$ and $$ 64 = 2^6 $$. Thus, we have:

$$
64 = (2^3)^{2} = 8^{2}
$$

Now we can write:

$$
\log_{8}(64) = \log_{8}(8^{2})
$$

Using the property of logarithms:

$$
\log_{8}(8^{2}) = 2 \cdot \log_{8}(8)
$$

Since $$ \log_{8}(8) = 1 $$, we have:

$$
\log_{8}(64) = 2 \cdot 1 = 2
$$

So, the answer is:

$$
\log_{8}(64) = 2
$$

In summary, the evaluations are:
a) $$ \frac{1}{2} $$
b) $$ -2 $$
c) $$ 2 $$
a) $$ \frac{1}{2} $$ b) $$ -2 $$ c) $$ 2 $$

Evaluate: a) \( \log _{19}(\sqrt{19})=\square \) b) \( \log _{4}\left(\frac{1}{16}\right)=\square \) c) \( \log _{8}(64)=\square \)