Question

Simplify 5 log2 k − 8 log2 m + 10 log2 n.

7 log2 (k − m + n)
7 log2 kn over m
log2 50 kn over 8 m
log2 k to the fifth power n to the tenth power over m to the eighth power

Answer 1

To simplify the expression \( 5 \log_2 k - 8 \log_2 m + 10 \log_2 n \), we can use the properties of logarithms.

1. Coefficient to exponent: \( a \log_b x = \log_b (x^a) \)
2. Logarithm of a quotient: \( \log_b (x) - \log_b (y) = \log_b \left(\frac{x}{y}\right) \)
3. Logarithm of a product: \( \log_b (x) + \log_b (y) = \log_b (xy) \)

Applying these properties:

1. Convert coefficients to exponents:
\[
5 \log_2 k = \log_2 (k^5)
\]
\[
-8 \log_2 m = -\log_2 (m^8)
\]
\[
10 \log_2 n = \log_2 (n^{10})
\]

2. Substitute these into the original expression:
\[
5 \log_2 k - 8 \log_2 m + 10 \log_2 n = \log_2 (k^5) - \log_2 (m^8) + \log_2 (n^{10})
\]

3. Combine the logarithms:
\[
= \log_2 (k^5) + \log_2 (n^{10}) - \log_2 (m^8)
\]
\[
= \log_2 \left(\frac{k^5 n^{10}}{m^8}\right)
\]

Thus, the simplified form of the expression is:
\[
\log_2 \left(\frac{k^5 n^{10}}{m^8}\right)
\]

The correct option from the choices you provided is:
\[
\log_2 \frac{k^5 n^{10}}{m^8}
\]