Solve (x^log10 3)^2-(3^log10 x)-2=0.

Name: Problem 9, Chapter 1: Mathematics for Joint Entrance Examination JEE (Advanced) Trigonometry
Uploaded: 2021-07-10T14:37:24-08:00
Duration: 3 min 25 s
Channel: Darshan Maheshwari
Description: Solve (x^log10 3)^2-(3^log10 x)-2=0.

Key Concepts

Exponential Expressions with Logarithmic Exponents

This concept involves expressions where an exponent is a logarithmic function. Understanding how to manipulate these expressions is key, especially recognizing that exponents and logarithms interact through identities which can simplify complex equations into more manageable forms.

Logarithm Properties

Logarithm properties, such as the power rule and change-of-base formula, allow one to manipulate and simplify expressions that contain logarithms. These properties enable the transformation of expressions with nested logarithms and exponents into equivalent forms that are easier to solve.

Change of Base Formula

The change-of-base formula is a fundamental tool in logarithms that allows one to convert logarithms from one base to another. This is particularly useful when dealing with expressions where the base of the logarithm does not match the base of the exponential term, facilitating the conversion to a common base.

Substitution Method for Solving Equations

This method involves replacing a complicated or repeated expression with a simpler variable to turn a complex equation into a standard form, such as a quadratic equation. Once the equation is simplified and solved, the substitution is reversed to find the original variable, providing a clearer pathway to the solution.

Transcript

0:00 Let's talk about question 24.

00:01 So we are given that log, logarithmic equation, and we need to find a value of x.

00:06 So the best way to isolate x and these in this case is to take antelog both sides.

00:12 So if we take antelog both sides, the log would just disappear and we would be left with the expression, which is inside the log, which will look something like this.

00:21 So let's move everything over to the left.

00:23 So we have 2x2x2x2x plus 3x minus 10x minus 30 is equal to 0.

00:30 Now this is going to be 2x squared and 3x minus 10x is nothing but minus 7x minus 30 is equal to 0.

00:37 Now this is a quadratic equation and we can try to factorize this.

00:41 So for that we first write the value of a, b and c.

00:43 So a is 2, b is minus 7 and c is minus 30.

00:49 So the value we multiply a and c to get negative 60.

00:53 And we write the factors of negative 60 in such a way that they sum up to give the value of b.

01:00 So let's see what can be the possible factors it can be six times minus 10 but they sum up to give us minus four that's not really something which we need uh let's see what can be the other option can it be four times um not really uh then what can be eight 12 12 times 5 yes i think we have a winner because 12 times 5 is 60 12 times 5 is 60 and negative 12 plus is negative 7 exactly what we need.

01:34 So we're going to replace the negative 7x, the negative 7x with negative 12x plus 5x.

01:42 These are two numbers in which we are interested in.

01:44 So negative 12x plus 5x minus 30 is equal to 0.

01:48 So we're going to take first pair and the second pair...

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