Question

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Answer 1

The expression of logarithm is proved with the help of exponent rule and logarithmic rules shown in the image.

$\log_b a=\dfrac{\log_x a}{log_xb}$

$\log_a(b^y)=y\log_a b$

What is change of base rule of logarithmic function?

The change of base rule of the log is used to write the given logarithmic number in terms of ratio of two log numbers.

For example,

$\log_b a=\dfrac{\log_x a}{log_xb}$

Here, a,b is the real number and x is the base.

The first expression have to be prove is,

$\log_a\dfrac{b}{c}=\log_ab-\log_ac$

For this, let suppose,

$a^x=b \;\; \text{and} \;\;a^y=c$

From the definition of the log,

$\log a^b=x \;\; \text{and} \;\;\log a^c=y$            

Now, by the rule of exponent,

$\dfrac{b}{c}=\dfrac{a^x}{a^y}=a^{x-y}$

Again, by the definition of the log,

$\log_a\dfrac{b}{c}=x-y\\\log_a\dfrac{b}{c}=\log_ab-\log_ac$

The second expression have to be prove is,

$\log_a(b^y)=y\log_a b$

For this, let suppose,

$b=a^x$

Take the left side of logarithmic expression, which has to be proved.

$\log_a(b^y)=\log_a (a^x)^y$            .....1

Now, by the rule of exponent,

$(a^x)^y=a^{xy}$

Again, by the equation 1,

$\log_a(b^y)=\log_a a^{xy}\\\log_a(b^y)={xy}\\\log_a(b^y)=y\log_a b$

Hence, the expression of logarithm is proved with the help of exponent rule and logarithmic rules shown in the image.

Learn more about the rules of logarithmic function here;

https://brainly.com/question/13473114