Show that log9 (xy2
) = ½ log3 x + log3 y
since 9 = 3^2, log3 x = 2 log9 x
so now we have
log9 xy^2 = log9 x + 2log9 y = 1/2 log3 x + log3 y
To show that log₉(xy²) = ½log₃x + log₃y, we can start by using the logarithmic identity:
logₐ(b⋅c) = logₐb + logₐc,
where a, b, and c are positive numbers.
Using this identity, we can rewrite log₉(xy²) as:
log₉(xy²) = log₉x + log₉(y²).
Next, we can use another logarithmic identity:
logₐ(bⁿ) = n⋅logₐb,
where a and b are positive numbers and n is a real number.
Applying this identity to log₉(y²), we can rewrite it as 2⋅log₉y.
Now, substituting these results back into log₉(xy²), we get:
log₉(xy²) = log₉x + 2⋅log₉y.
To simplify this expression further, we can use the change of base formula:
logₐb = logₙb / logₙa,
where a > 0, b > 0, and n > 0.
Applying this formula with base 9, we have:
log₉x = log₃x / log₃9,
and
log₉y = log₃y / log₃9.
Since log₃9 = 2, we can substitute these expressions back into log₉(xy²):
log₉(xy²) = log₃x / log₃9 + 2⋅log₃y / log₃9.
Now, to simplify further, we divide both terms by log₃9:
log₉(xy²) = (log₃x) / 2 + 2(log₃y) / 2.
Simplifying this expression gives us:
log₉(xy²) = ½log₃x + log₃y.
Therefore, we have shown that log₉(xy²) = ½log₃x + log₃y.
To prove the given equation:
log9 (xy^2) = 1/2 * log3 x + log3 y
We can start by expressing each term in terms of a common logarithm base, such as log10.
Using the change of base formula, we can rewrite the equation as:
log9 (xy^2) = log(x) / log(9) + 1/2 * log(x) / log(3) + log(y) / log(3)
Now, let's simplify each term individually:
First term:
log9 (xy^2) = log(xy^2) / log(9) (Property of logarithms)
Second term:
log(x) / log(9) = log(x) / log(3^2) (9 can be written as 3^2)
= log(x) / (2 * log(3)) (Property of logarithms)
= 1/2 * log(x) / log(3)
Third term:
log(y) / log(3) = log(y) / log(3) (No further simplification needed)
Now, let's substitute the simplified terms back into the equation:
log(xy^2) / log(9) = 1/2 * log(x) / log(3) + log(y) / log(3)
Multiplying both sides by log(9), we get:
log(xy^2) = 1/2 * log(x) * log(9) / log(3) + log(y) * log(9) / log(3)
Since log(a * b) = log(a) + log(b), the equation becomes:
log(xy^2) = 1/2 * log(x) + log(9) / log(3) + log(y) + log(9) / log(3)
Now, remember that log(9) = log(3^2) = 2 * log(3). Substituting this in, we have:
log(xy^2) = 1/2 * log(x) + 2 * log(3) / log(3) + log(y) + 2 * log(3) / log(3)
Simplifying further:
log(xy^2) = 1/2 * log(x) + 2 + log(y) + 2
log(xy^2) = 1/2 * log(x) + log(y) + 4
Thus, we've proved that log9 (xy^2) = 1/2 * log3 x + log3 y.