For Problems 5 through 16, first find the value of the logarithms without using a calculator. Ch

For Problems 5 through 16, first find the value of the...

Key Concepts

Logarithm

A logarithm is the inverse operation of exponentiation; it finds the power to which a given base must be raised to produce a specific number. This concept is fundamental in solving equations involving exponential growth or decay and in simplifying complex multiplicative relationships.

Common Logarithm

A common logarithm specifically uses base 10. It is frequently used in a variety of real-world contexts because our number system is decimal, and it simplifies the computation of orders of magnitude in science and engineering.

Properties of Logarithms

Logarithms possess several useful properties, such as the product rule (log(ab) = log a + log b), the quotient rule (log(a/b) = log a - log b), and the power rule (log(a^b) = b log a). These properties allow for the breakdown of complex logarithmic expressions into simpler parts that can be evaluated manually without a calculator.

Exponential Functions

Exponential functions, which represent repeated multiplication by a base, are closely related to logarithms as they serve as their inverses. Understanding exponential functions helps in comprehending how logarithms work and why they are used to solve equations involving exponential expressions.

Transcript

00:01 We are given with log base 0 .5 of 16 and we're supposed to find the value of it without using a calculator.

00:12 So let's just say that y is equal to log base 0 .5 of 16 and this normal procedure is to compare it with the standard equation that means y is equal to log base b of x.

00:30 This means that y is the power of b so then we'll get x that is the exponential form of this is x is equal to b to the power of y but the thing here is the base we have is 0 .5 so it's not easy for us to calculate or find the value of this one without using a catheter so if we just do just a simple thing you can just solve it easily so it's like instead of writing point five what we have to do is that which is right y is equal to log point five is nothing but one by two right one by two that is a basis one by two of 16 and now they can compare the equations okay so if you have a decimal value here then just convert into a fraction and then do the problem okay or else it will be really difficult for you if you do this it will be really easy i'll show you that so when we compare these two, instead of y, x and b, we have, instead of y, we have, instead of y, we have y itself.

01:48 Instead of x, we have 16, and instead of b, we have 1 by 2.

01:54 So when we write the exponential equal of this equation, we'll get 16 is equal to 1 by 2, the power of one the one thing you need to note is that if aether of these two are fractions then here you will get negative that is if one of them is fraction then you will get negative if two of them are fractions then you may get a negative but i'm not sure that about that you will have to do the problem then only can find out but if one of the misnick there is fraction then you will get bias next to right so that you can just check with your answer i mean if it's right or not so here it's like you need to, you have two and you have 16 here.

02:42 So you have to rate somehow these two...

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