When determining if you can solve an exponential equation, certain conditions must be satisfied:
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Base Is Positive: The base of the exponential functions must be positive (greater than 0) and not equal to 1, because if the base is 1, the function is constant and does not yield distinct solutions.
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Equal Bases: If possible, you should aim to express both sides of the equation with the same base. This allows you to use the property that if \( a^x = a^y \), then \( x = y \).
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Logarithmic Approach: If the bases cannot be made the same, you can still solve the equation by taking the logarithm of both sides. For example, if you have an equation of the form \( a^x = b \), you can take the logarithm (natural log, base 10, etc.) of both sides and use the property of logarithms: \( x \log(a) = \log(b) \).
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Undefined Cases: If the exponential equation involves values that could lead to undefined scenarios (like taking the logarithm of a negative number or zero), then those specific cases must be handled carefully or excluded.
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Consider Domain Restrictions: Ensure that the variables involved give defined outputs within the domain of the function (for example, if you have \( e^x \), the output should be positive).
By satisfying these conditions, you should be able to determine whether you can solve a given exponential equation and how to approach it.