Question

Доказать что a^(lgb)=b^(lga)

Answer 1

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Answer 2

a^(lgb); прологарифмируем по основанию 10.

lgb*lga=lga*lgb=lg(b^(lga))=lg(a^(lgb))

b^(lga)=a^(lgb)

Answer 3

To prove that a^(lgb) = b^(lga), we can take the logarithm of both sides using any base. Let's take the natural logarithm (ln) for convenience:

ln(a^(lgb)) = ln(b^(lga))

Using the logarithmic property that ln(x^y) = y * ln(x), we can simplify the left and right sides of the equation:

lgb * ln(a) = lga * ln(b)

Now we can divide both sides by lga * lnb:

ln(a)/ln(b) = lga/lgb

We know that loga(b) = ln(b)/ln(a) (by the change of base formula), so we can substitute that into the equation:

loga(b) = lga/lgb

Multiplying both sides by lgb gives:

lgb * loga(b) = lga

And since we know that logb(a) = 1/loga(b), we can substitute that in:

lgb / logb(a) = lga

Multiplying both sides by logb(a) gives:

lgb = lga * logb(a)

And finally, substituting this back into the original equation gives:

a^(lga * logb(a)) = b^(lgb)

Using the power rule of logarithms, we can simplify the left side:

(a^logb(a))^lga = b^(lgb)

Since a^logb(a) = b^loga(b) = b, we get:

b^lga = b^lgb

And therefore, a^(lgb) = b^(lga), which is what we wanted to prove.