How a^{\log_b x} = x^{\log_b a}

tefoldaavv

tefoldaavv

Answered question

2022-04-26

How alogbx=xlogba

Answer & Explanation

Leia Wiggins

Leia Wiggins

Beginner2022-04-27Added 18 answers

For any positive r and you have 
rs=blogb(rs)=bslogb(r).
So, taking r=a and s=logb(x), we have:
alogb(x)=blogb(x)logb(a)
=blogb(a)logb(x)
=blogb(xlogb(a))
=xlogb(a).

Gianna Travis

Gianna Travis

Beginner2022-04-28Added 11 answers

Using log properties, we have
alogb(x)=blogb(alogb(x)}=blogb(x)logb(a)=(blogb(x))logb(a)=xlogb(a)

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