Write the expression below so that only a single logarithm or exponential function appears.. . (1/2)ln(z) - ln(5 + x) - 4 ln(y)

Solution:

Given, the expression is (1/2)ln(z) - ln(5 + x) - 4 ln(y).

We have to write the expression in such a way that only a single logarithm or exponential function appears.

By using logarithmic property,

ln(a^b) = b.ln(a)

Now, (1/2)ln(z) = ln(z^1/2)

Similarly, 4ln(y) = ln(y⁴)

(1/2)ln(z) - ln(5 + x) - 4 ln(y) = ln(z^1/2) - ln(5 + x) - ln(y⁴)

By using logarithmic property,

ln(a) + ln(b) = ln(a.b)

ln(z^1/2) - ln(5 + x) - ln(y⁴) = ln(z^1/2) - [ln(5 + x) + ln(y⁴)]

= ln(z^1/2) - ln[(5 + x).(y⁴)]

By using logarithmic property,

ln(a) - ln(b) = ln(a/b)

ln(z^1/2) - ln[(5 + x).(y⁴)] = ln[(z^1/2) / (5 + x).(y⁴)]

Therefore, the required solution is ln[(z^1/2) / (5 + x).(y⁴)]

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Write the expression below so that only a single logarithm or exponential function appears.. . (1/2)ln(z) - ln(5 + x) - 4 ln(y)

Summary:

The expression (1/2)ln(z) - ln(5 + x) - 4 ln(y) with only a single logarithm or exponential function is ln[(z^1/2) / (5 + x).(y⁴)]