SOLUTION: Express in a single logarithm, ln x + a ln y − b ln z

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Question 1153449: Express in a single logarithm, ln x + a ln y − b ln z
Found 3 solutions by ikleyn, Theo, josmiceli:
Answer by ikleyn(52794)   (Show Source):
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On logarithms and their properties, see introductory lessons
    - WHAT IS the logarithm
    - Properties of the logarithm
    - Change of Base Formula for logarithms
in this site.

Also, you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic "Logarithms".

Save the link to this online textbook together with its description

Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson

to your archive and use it when it is needed.

Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website!
two properties of logs are applicable here.

first is :

ln(x^a) = a * ln(x)

second is:

ln(x * y / z) = ln(x) * ln(y) - ln(z)

first gets you:

a * ln(y) = ln(y^a)

b * ln(z) = ln(z^b)

your formula starts with:

ln(x) + a * ln(y) - b * ln(z)

applying the first property above makes the expression above equivalent to:

ln(x) + ln(y^a) - ln(z^b)

second gets you:

ln(x * y^a / z^b) = ln(x) + ln(y^a) - ln(z^b)

here's a reference on properties of logarithms.

the log log function in your calculator is log(x) to the base of 10.

the ln function is log(x) in your calculator is log(x) to the base of e.

the properties of logs applies to both log function and ln function.

here's a couple of references on the properties of logs.

http://www.montereyinstitute.org/courses/DevelopmentalMath/TEXTGROUP-1-19_RESOURCE/U18_L2_T2_text_final.html

https://www.purplemath.com/modules/logrules.htm

Answer by josmiceli(19441)   (Show Source):
You can put this solution on YOUR website!