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Question: How do you write \({b^x} = y\) into logarithmic form?...

How do you write bx=y{b^x} = y into logarithmic form?

Explanation

Solution

We will first write the given expression and then take logarithmic function on both the sides of the given equation and then use the property that log⁡ab=blog⁡a\log {a^b} = b\log a.

Complete step-by-step answer:
We are given that we are required to write bx=y{b^x} = y into logarithmic form.
Let us assume bx=y{b^x} = y to be equation number 1.
Now, taking logarithmic function on both the sides if equation number 1, we will then obtain the following expression:-
⇒log⁡bx=log⁡y\Rightarrow \log {b^x} = \log y ……………..(2)
Now we will use the property of logarithmic which states that log⁡ab=blog⁡a\log {a^b} = b\log a.
Replacing a by b and b by x in the above mentioned property of logarithmic function, we will then obtain the following equation:-
⇒log⁡bx=xlog⁡b\Rightarrow \log {b^x} = x\log b
Putting this in equation number 2, we will then obtain the following expression:-
⇒xlog⁡b=log⁡y\Rightarrow x\log b = \log y
Now, we will take the log b from multiplication in the left hand side to division in the right hand side, we will then obtain the following equation:-
⇒x=log⁡ylog⁡b\Rightarrow x = \dfrac{{\log y}}{{\log b}} …………….(3)
Now, we use the property of logarithmic which states that log⁡alog⁡b=log⁡ba\dfrac{{\log a}}{{\log b}} = {\log _b}a.
Replacing a by y and b by nothing in the above mentioned property of logarithmic function, we will then obtain the following equation:-
⇒log⁡ylog⁡b=log⁡by\Rightarrow \dfrac{{\log y}}{{\log b}} = {\log _b}y
Putting this in equation number 3, we will then obtain the following expression:-

**⇒x=log⁡by \Rightarrow x = {\log _b}y
Thus, we have the required answer. **

Note:
The students must commit to memory the following properties and formulas related to the logarithmic functions:
log⁡ab=blog⁡a\log {a^b} = b\log a
log⁡alog⁡b=log⁡ba\dfrac{{\log a}}{{\log b}} = {\log _b}a
The students must also notice the fact that we ourselves introduced the logarithmic function, it was nowhere given in the question and then we just modified it to present it in a better way. Always keep this in mind, whenever required to convert any function into logarithmic, just take log on both the sides of the given expression and then just keep on modifying it using the properties of logarithmic function.