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Hello. Welcome to We Learn Daily
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In this video, I'll show you how to solve an equation using powers and logarithms
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Be sure to watch until the end and let me know in the comments if the solution was clear
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and which part you found most interesting. Let's start. Notice that the numbers 9 and 27 are powers of 3
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This means that the equation can be rewritten using powers of 3
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The term three raised to the power of x remains the same. The term nine raised to the power of x can be rewritten as three raised to the power of 2
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all raised to the power of x, and the term 27 raised to the power of x can be rewritten as three raised to the power of three
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all raised to the power of x. When raising a power to another power, the exponents are multiplied
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This allows us to rewrite the equation as three raised to the power of x
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plus three raised to the power of two times x plus 3 raised to the power of 3 times x equals 14 Next it convenient to express the second and third terms as powers with the base 3 raised to the power of x
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So the first term remains the same, while the second and third terms can be written as 3 raised to the power of x squared
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and 3 raised to the power of x cubed. This gives us an equation that depends on 3 raised to the power of x
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So we'll let t represent 3 raised to the power of x
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The equation now becomes t plus t squared plus t cubed equals 14
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Next, we move all terms to one side of the equation to set it equal to 0
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t cubed plus t squared plus t minus 14 equals 0. To solve this cubic equation, we can look for potential roots
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One of the roots can be found among the divisors of the constant term, which is 2
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This means that the left side of the equation can be factored. with one of the factors being t minus 2
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Let try to isolate this factor on the left side To do this we subtract 2t squared but to keep the equation balanced we add 2t squared In the difference t cubed minus t squared we can factor out t squared
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We get t squared multiplied by t minus 2, and we still have 3t squared plus t minus 14, all of which equals 0
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Next, we subtract 6t, but to keep the equation balanced, we add 6 to the left side of the equation
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We then rewrite the first term. For the next pair, we factor out 3t
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As a result, we get 3t multiplied by t minus 2, and we still have plus 7t minus 14
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We rewrite the first two terms. From the last pair, we can factor out the number 7
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resulting in plus 7 multiplied by t minus 2, and all of this equals 0
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Finally, we factor out t minus 2, leaving us with t squared plus 3t plus 7
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And all of this equals 0 The product of two polynomials equals 0 when at least one of them is equal to 0 This means either t minus 2 equals 0 or t squared plus 3 t plus 7 equals 0 It is immediately
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noticeable that the discriminant of the quadratic trinomial on the left side of the second equation
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is less than 0, which means the second equation has no real roots. Therefore, the variable t
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takes only one value, which is 2. Returning to our substitution, we get 3 raised to the power of x equals 2
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The equation has a single root where the value of x is equal to the logarithm of 2 to the base 3
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For those who may have forgotten the definition of a logarithm, you can take the logarithm of both sides of the last equation with base 3
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The result will be the same. The equation has a single root where the value of x is equal to the logarithm of 2 to the base 3
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The problem is solved. If the solution is clear, please like the video, leave a comment, and don't forget to subscribe to the channel
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Thank you for watching and see you in the next video. Goodbye
